{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0001","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d25aff81e3c26a24d9ac9544f1f5a924725db74c51b93e4bcf22fe63fb96f4a","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056001,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,8],[2,8],[3,8],[2,8]],[[2,12],[3,12],[4,12],[3,12]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,1,2]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/8], [1, 3/8], [2, 1/4]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/8], [1, 3/8], [2, 1/4]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0002","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eaf247c3ec7cc4fc280a1715988c35d6996e07c52207b1c6c21a29b9c4e445d","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056002,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,9],[2,9],[3,9],[3,9]],[[2,13],[3,13],[4,13],[4,13]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,2,3]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/9, 2/9, 3/9, 3/9], [2/13, 3/13, 4/13, 4/13]], and one total parameter-independent statistic T=[0, 0, 2, 3] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/13], [2, 4/13], [3, 4/13]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/13], [2, 4/13], [3, 4/13]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0003","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ae2eb177edae2ca93d152129dd122eb0b3cd1a73b5660d4c2b52a9e8e27ea2a","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056003,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,3,4]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one total parameter-independent statistic T=[0, 0, 3, 4] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/10], [3, 3/10], [4, 2/5]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/10], [3, 3/10], [4, 2/5]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0004","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e09a391cbc4bc286312af2b9de79f29ffd66e6f6837fe1581cb107b9959107a","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056004,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,4,5]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one total parameter-independent statistic T=[0, 0, 4, 5] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 1/3], [4, 4/15], [5, 2/5]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 1/3], [4, 4/15], [5, 2/5]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0005","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2103dcef6fc439cae25c2f35eb2d90535e3b8e3560dbc3b68bb9a3274b00dcb9","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056005,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,5,6]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one total parameter-independent statistic T=[0, 0, 5, 6] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 1/4], [5, 1/4], [6, 1/2]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 1/4], [5, 1/4], [6, 1/2]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0006","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07ddd5b2dea01e9ae95c05cd6b957fc63eae25982cfc7497e8ce8529b45107a6","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056006,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,6,7]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one total parameter-independent statistic T=[0, 0, 6, 7] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/17], [6, 4/17], [7, 8/17]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/17], [6, 4/17], [7, 8/17]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0007","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a55fb86392b6f67b103b0c4b6d72d73949f738276d16304fd694274a812101c","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056007,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,7,8]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one total parameter-independent statistic T=[0, 0, 7, 8] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/14], [7, 3/14], [8, 4/7]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/14], [7, 3/14], [8, 4/7]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0008","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4b07ba36dcbb18dba0fe8db5030cc43cfaa81ec820a59ecc0fc3e1ae9d173d","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056008,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,8,9]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one total parameter-independent statistic T=[0, 0, 8, 9] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/19], [8, 4/19], [9, 10/19]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/19], [8, 4/19], [9, 10/19]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0009","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20069f699715c4e37794ab3d6d27386591c5efab5dff5d544fa47eef03330fe2","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056009,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,9,10]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one total parameter-independent statistic T=[0, 0, 9, 10] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/16], [9, 3/16], [10, 5/8]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/16], [9, 3/16], [10, 5/8]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0010","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce59d7b7519417f9e7f8b862540ec0120eb99dec3dcda7fe33703a55a6b0d83","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056010,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,10,11]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one total parameter-independent statistic T=[0, 0, 10, 11] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/21], [10, 4/21], [11, 4/7]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/21], [10, 4/21], [11, 4/7]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0011","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab40456a85b37d65419695869050524f107307e3fa1cd52768d27272ff4f808","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056011,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,11,12]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one total parameter-independent statistic T=[0, 0, 11, 12] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 1/6], [11, 1/6], [12, 2/3]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 1/6], [11, 1/6], [12, 2/3]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0012","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0154e75fc39e5dbb62de8ca7615fbae9c0971630b92d8efdced5dac515a04224","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056012,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,12,13]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one total parameter-independent statistic T=[0, 0, 12, 13] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/23], [12, 4/23], [13, 14/23]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/23], [12, 4/23], [13, 14/23]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0013","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f881f63d1bb31c92012eb45392ea6849b7d34e172cfe2b61ea3598c429bd8f19","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056013,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,13,14]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one total parameter-independent statistic T=[0, 0, 13, 14] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/20], [13, 3/20], [14, 7/10]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/20], [13, 3/20], [14, 7/10]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0014","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"380e0583f33e9e9d1a0fce69e0986af816a49d9ba89402bb6df5aba3f57f75ce","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056014,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,14,15]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one total parameter-independent statistic T=[0, 0, 14, 15] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 1/5], [14, 4/25], [15, 16/25]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 1/5], [14, 4/25], [15, 16/25]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0015","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b29a9cdfc992069c3ec305b7c272f8119fab6aa2c35715211ea344e53b4cdc53","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056015,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,15,16]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one total parameter-independent statistic T=[0, 0, 15, 16] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/22], [15, 3/22], [16, 8/11]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/22], [15, 3/22], [16, 8/11]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0016","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd99a9e908980543889f69c60f224710db628ddda700bb5aa1fea5952e97bd7a","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056016,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,16,17]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one total parameter-independent statistic T=[0, 0, 16, 17] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/27], [16, 4/27], [17, 2/3]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/27], [16, 4/27], [17, 2/3]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0017","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72e92992bfd771cb86c2b5f6fba81c6a173fed859ed5cd21efbf6d000d42b2a8","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056017,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,17,18]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one total parameter-independent statistic T=[0, 0, 17, 18] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 1/8], [17, 1/8], [18, 3/4]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 1/8], [17, 1/8], [18, 3/4]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0018","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da977d82ecf911342ee0692fb479537a7413ae9538580be700a99e0466538150","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056018,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,18,19]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one total parameter-independent statistic T=[0, 0, 18, 19] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/29], [18, 4/29], [19, 20/29]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/29], [18, 4/29], [19, 20/29]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0019","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d24aea35cd158a67aee36afd2fd43358a7710e170212053c173a8dea07e22f1d","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056019,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,19,20]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one total parameter-independent statistic T=[0, 0, 19, 20] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/26], [19, 3/26], [20, 10/13]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/26], [19, 3/26], [20, 10/13]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0020","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65461b37d0855aa2e156ac8f3a2f94791ed2629b8e046d1d61a8ea42520724d","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056020,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,20,21]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one total parameter-independent statistic T=[0, 0, 20, 21] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/31], [20, 4/31], [21, 22/31]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/31], [20, 4/31], [21, 22/31]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0021","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b3d468bb570c88259f7cbfac4cabf661a1448e2abb0bc4905b4c10930c5e155","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056021,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,21,22]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one total parameter-independent statistic T=[0, 0, 21, 22] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/28], [21, 3/28], [22, 11/14]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/28], [21, 3/28], [22, 11/14]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0022","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b1fe23a388e43afeae2e1ee4681facbd77d0c7d316667524b4f019582bc35ca","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056022,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,22,23]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one total parameter-independent statistic T=[0, 0, 22, 23] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 5/33], [22, 4/33], [23, 8/11]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 5/33], [22, 4/33], [23, 8/11]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0023","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"380bcead36f463dae7696c2f6024c2fbd41383b9d6fc132c31d28dd99b1ed94a","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056023,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,23,24]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one total parameter-independent statistic T=[0, 0, 23, 24] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 1/10], [23, 1/10], [24, 4/5]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 1/10], [23, 1/10], [24, 4/5]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0024","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55358b22b5f69c6932726e49e369da91a7ec918e88fdc53aa145b1f0336053f8","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056024,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,24,25]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one total parameter-independent statistic T=[0, 0, 24, 25] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(1, [[0, 1/7], [24, 4/35], [25, 26/35]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (1, [[0, 1/7], [24, 4/35], [25, 26/35]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0025","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0fd261f0337280e04b6bdf29c149c7d0b166152e9693fdbaa69c7561d60f396","family_id":"qf-mod36-sampling-pushforward-at-parameter","seed":202611056025,"semantic_template":"qf-mod36-sampling-pushforward-at-parameter/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,25,26]}"},"fields":{"title":"Sum Every Estimator Fibre under One Parameter Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one total parameter-independent statistic T=[0, 0, 25, 26] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0. Give (parameter, law, total_mass, model_qualified, certificate).","answer":{"kind":"expression","value":"(0, [[0, 3/32], [25, 3/32], [26, 13/16]], 1, Yes, pushforward_sum_over_estimator_fibres)","explanation":"Return the increasing list of statistic-value/mass pairs, including zero entries, and its exact total mass. This pushforward is the sampling distribution conditional on the supplied parameter, not a posterior or a finite-data estimate. The exact result, retaining the requested order, is (0, [[0, 3/32], [25, 3/32], [26, 13/16]], 1, Yes, pushforward_sum_over_estimator_fibres)."},"skills":["Sum Every Estimator Fibre under One Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0026","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5a5ce0aefc2cead973bb8211dbaf8bf52c2e2767063999de4d636a3a7da5ff4","family_id":"qf-mod36-sampling-family-readback","seed":202611056026,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,8],[2,8],[3,8],[2,8]],[[2,12],[3,12],[4,12],[3,12]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,1,2]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 1, 2], [[0, [[0, 3/8], [1, 3/8], [2, 1/4]]], [1, [[0, 5/12], [1, 1/3], [2, 1/4]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 1, 2], [[0, [[0, 3/8], [1, 3/8], [2, 1/4]]], [1, [[0, 5/12], [1, 1/3], [2, 1/4]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0027","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abdf4c5858369ae4892f63677a0cde3074a923c357ce7bca4a21d1667c6cfec8","family_id":"qf-mod36-sampling-family-readback","seed":202611056027,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,9],[2,9],[3,9],[3,9]],[[2,13],[3,13],[4,13],[4,13]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,2,3]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/9, 2/9, 3/9, 3/9], [2/13, 3/13, 4/13, 4/13]], and one total parameter-independent statistic T=[0, 0, 2, 3] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 2, 3], [[0, [[0, 1/3], [2, 1/3], [3, 1/3]]], [1, [[0, 5/13], [2, 4/13], [3, 4/13]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 2, 3], [[0, [[0, 1/3], [2, 1/3], [3, 1/3]]], [1, [[0, 5/13], [2, 4/13], [3, 4/13]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0028","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2855579d68e1af2ddb63ec9e00cd0b94e97838a97e858de264a8e05c3f7e4f16","family_id":"qf-mod36-sampling-family-readback","seed":202611056028,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,3,4]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one total parameter-independent statistic T=[0, 0, 3, 4] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 3, 4], [[0, [[0, 3/10], [3, 3/10], [4, 2/5]]], [1, [[0, 5/14], [3, 2/7], [4, 5/14]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 3, 4], [[0, [[0, 3/10], [3, 3/10], [4, 2/5]]], [1, [[0, 5/14], [3, 2/7], [4, 5/14]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0029","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8dbe8adf2bb467d76edc979563d4b56f180c60d646fb36e797540e8e0ddc52c","family_id":"qf-mod36-sampling-family-readback","seed":202611056029,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,4,5]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one total parameter-independent statistic T=[0, 0, 4, 5] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 4, 5], [[0, [[0, 3/11], [4, 3/11], [5, 5/11]]], [1, [[0, 1/3], [4, 4/15], [5, 2/5]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 4, 5], [[0, [[0, 3/11], [4, 3/11], [5, 5/11]]], [1, [[0, 1/3], [4, 4/15], [5, 2/5]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0030","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b062284fcdc92641ee8e4d5b997b1b65b67686ed121baa91e38cc6d0b7f0ee57","family_id":"qf-mod36-sampling-family-readback","seed":202611056030,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,5,6]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one total parameter-independent statistic T=[0, 0, 5, 6] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 5, 6], [[0, [[0, 1/4], [5, 1/4], [6, 1/2]]], [1, [[0, 5/16], [5, 1/4], [6, 7/16]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 5, 6], [[0, [[0, 1/4], [5, 1/4], [6, 1/2]]], [1, [[0, 5/16], [5, 1/4], [6, 7/16]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0031","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b484dfb3eab7aee6ac6e633b4db70351d3b944595179e8a1ea1597201601ece","family_id":"qf-mod36-sampling-family-readback","seed":202611056031,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,6,7]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one total parameter-independent statistic T=[0, 0, 6, 7] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 6, 7], [[0, [[0, 3/13], [6, 3/13], [7, 7/13]]], [1, [[0, 5/17], [6, 4/17], [7, 8/17]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 6, 7], [[0, [[0, 3/13], [6, 3/13], [7, 7/13]]], [1, [[0, 5/17], [6, 4/17], [7, 8/17]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0032","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"854f8675b0e238ee29f360ceda0616d20b9a6c79c20c04eb3b163f61d71c6fa2","family_id":"qf-mod36-sampling-family-readback","seed":202611056032,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,7,8]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one total parameter-independent statistic T=[0, 0, 7, 8] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 7, 8], [[0, [[0, 3/14], [7, 3/14], [8, 4/7]]], [1, [[0, 5/18], [7, 2/9], [8, 1/2]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 7, 8], [[0, [[0, 3/14], [7, 3/14], [8, 4/7]]], [1, [[0, 5/18], [7, 2/9], [8, 1/2]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0033","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d4e3f975471dd304309a748bc2c7d50911c1b433a409869dcb7e9f4847c41f","family_id":"qf-mod36-sampling-family-readback","seed":202611056033,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,8,9]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one total parameter-independent statistic T=[0, 0, 8, 9] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 8, 9], [[0, [[0, 1/5], [8, 1/5], [9, 3/5]]], [1, [[0, 5/19], [8, 4/19], [9, 10/19]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 8, 9], [[0, [[0, 1/5], [8, 1/5], [9, 3/5]]], [1, [[0, 5/19], [8, 4/19], [9, 10/19]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0034","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"223348436205dda5380b5091251dac413be7a719e425708004f52c3ac50ce145","family_id":"qf-mod36-sampling-family-readback","seed":202611056034,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,9,10]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one total parameter-independent statistic T=[0, 0, 9, 10] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 9, 10], [[0, [[0, 3/16], [9, 3/16], [10, 5/8]]], [1, [[0, 1/4], [9, 1/5], [10, 11/20]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 9, 10], [[0, [[0, 3/16], [9, 3/16], [10, 5/8]]], [1, [[0, 1/4], [9, 1/5], [10, 11/20]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0035","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69005276948d66af9d99c1ca649b393919f7399913064ede5efaae66c132eec7","family_id":"qf-mod36-sampling-family-readback","seed":202611056035,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,10,11]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one total parameter-independent statistic T=[0, 0, 10, 11] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 10, 11], [[0, [[0, 3/17], [10, 3/17], [11, 11/17]]], [1, [[0, 5/21], [10, 4/21], [11, 4/7]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 10, 11], [[0, [[0, 3/17], [10, 3/17], [11, 11/17]]], [1, [[0, 5/21], [10, 4/21], [11, 4/7]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0036","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8644e175125468dd9036ba1f42d1998a260827165caf50ab3a4ffc1aa9bf272f","family_id":"qf-mod36-sampling-family-readback","seed":202611056036,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,11,12]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one total parameter-independent statistic T=[0, 0, 11, 12] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 11, 12], [[0, [[0, 1/6], [11, 1/6], [12, 2/3]]], [1, [[0, 5/22], [11, 2/11], [12, 13/22]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 11, 12], [[0, [[0, 1/6], [11, 1/6], [12, 2/3]]], [1, [[0, 5/22], [11, 2/11], [12, 13/22]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0037","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a32bb8d4b58b3895952938dc976362a1dc126bd4a7986716d29c5330efa9334f","family_id":"qf-mod36-sampling-family-readback","seed":202611056037,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,12,13]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one total parameter-independent statistic T=[0, 0, 12, 13] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 12, 13], [[0, [[0, 3/19], [12, 3/19], [13, 13/19]]], [1, [[0, 5/23], [12, 4/23], [13, 14/23]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 12, 13], [[0, [[0, 3/19], [12, 3/19], [13, 13/19]]], [1, [[0, 5/23], [12, 4/23], [13, 14/23]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0038","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dff34f84f67749a1bfc4896d8c986ff89c24f40cbdc7fc8ff2d641f9a5e92216","family_id":"qf-mod36-sampling-family-readback","seed":202611056038,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,13,14]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one total parameter-independent statistic T=[0, 0, 13, 14] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 13, 14], [[0, [[0, 3/20], [13, 3/20], [14, 7/10]]], [1, [[0, 5/24], [13, 1/6], [14, 5/8]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 13, 14], [[0, [[0, 3/20], [13, 3/20], [14, 7/10]]], [1, [[0, 5/24], [13, 1/6], [14, 5/8]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0039","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3459aeebd13654d3f75ef9698baef642674635290268715735c1a5116db9807d","family_id":"qf-mod36-sampling-family-readback","seed":202611056039,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,14,15]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one total parameter-independent statistic T=[0, 0, 14, 15] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 14, 15], [[0, [[0, 1/7], [14, 1/7], [15, 5/7]]], [1, [[0, 1/5], [14, 4/25], [15, 16/25]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 14, 15], [[0, [[0, 1/7], [14, 1/7], [15, 5/7]]], [1, [[0, 1/5], [14, 4/25], [15, 16/25]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0040","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d81c1f78e4e6d2445abeb5bf6b83be964439be7e200a107fceb96c967b6111d9","family_id":"qf-mod36-sampling-family-readback","seed":202611056040,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,15,16]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one total parameter-independent statistic T=[0, 0, 15, 16] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 15, 16], [[0, [[0, 3/22], [15, 3/22], [16, 8/11]]], [1, [[0, 5/26], [15, 2/13], [16, 17/26]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 15, 16], [[0, [[0, 3/22], [15, 3/22], [16, 8/11]]], [1, [[0, 5/26], [15, 2/13], [16, 17/26]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0041","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eead72eb14483007af8394dbd4247edd0b22b8d1f00051caea0b6c98ba91b618","family_id":"qf-mod36-sampling-family-readback","seed":202611056041,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,16,17]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one total parameter-independent statistic T=[0, 0, 16, 17] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 16, 17], [[0, [[0, 3/23], [16, 3/23], [17, 17/23]]], [1, [[0, 5/27], [16, 4/27], [17, 2/3]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 16, 17], [[0, [[0, 3/23], [16, 3/23], [17, 17/23]]], [1, [[0, 5/27], [16, 4/27], [17, 2/3]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0042","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b08d9d7397d2b9249129126abeef8d19c27f338c8056f2419bc83fc71e139c83","family_id":"qf-mod36-sampling-family-readback","seed":202611056042,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,17,18]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one total parameter-independent statistic T=[0, 0, 17, 18] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 17, 18], [[0, [[0, 1/8], [17, 1/8], [18, 3/4]]], [1, [[0, 5/28], [17, 1/7], [18, 19/28]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 17, 18], [[0, [[0, 1/8], [17, 1/8], [18, 3/4]]], [1, [[0, 5/28], [17, 1/7], [18, 19/28]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0043","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b81f0eeb17a110b2719b2c96915dc154830ed67d7b8648fc3c23a822809e479","family_id":"qf-mod36-sampling-family-readback","seed":202611056043,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,18,19]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one total parameter-independent statistic T=[0, 0, 18, 19] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 18, 19], [[0, [[0, 3/25], [18, 3/25], [19, 19/25]]], [1, [[0, 5/29], [18, 4/29], [19, 20/29]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 18, 19], [[0, [[0, 3/25], [18, 3/25], [19, 19/25]]], [1, [[0, 5/29], [18, 4/29], [19, 20/29]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0044","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f11f0ff5eb89bddc73938eb675f2f6a91f4b46570b36ef6e868d224d98f67ec7","family_id":"qf-mod36-sampling-family-readback","seed":202611056044,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,19,20]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one total parameter-independent statistic T=[0, 0, 19, 20] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 19, 20], [[0, [[0, 3/26], [19, 3/26], [20, 10/13]]], [1, [[0, 1/6], [19, 2/15], [20, 7/10]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 19, 20], [[0, [[0, 3/26], [19, 3/26], [20, 10/13]]], [1, [[0, 1/6], [19, 2/15], [20, 7/10]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0045","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ea4f265b7932ce5a5605342193fc1d60768c692a8f0739133ea7324b422637d","family_id":"qf-mod36-sampling-family-readback","seed":202611056045,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,20,21]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one total parameter-independent statistic T=[0, 0, 20, 21] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 20, 21], [[0, [[0, 1/9], [20, 1/9], [21, 7/9]]], [1, [[0, 5/31], [20, 4/31], [21, 22/31]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 20, 21], [[0, [[0, 1/9], [20, 1/9], [21, 7/9]]], [1, [[0, 5/31], [20, 4/31], [21, 22/31]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0046","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75bd8beef88f4c34b50c8847a6a3f36fdd23d1b24e6c725c6e5dd0a28fff45ce","family_id":"qf-mod36-sampling-family-readback","seed":202611056046,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,21,22]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one total parameter-independent statistic T=[0, 0, 21, 22] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 21, 22], [[0, [[0, 3/28], [21, 3/28], [22, 11/14]]], [1, [[0, 5/32], [21, 1/8], [22, 23/32]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 21, 22], [[0, [[0, 3/28], [21, 3/28], [22, 11/14]]], [1, [[0, 5/32], [21, 1/8], [22, 23/32]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0047","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49cf9c66db7bb09043db9ca9d8afe1b20ee255656fb2b1d1445be41b4cced9d8","family_id":"qf-mod36-sampling-family-readback","seed":202611056047,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,22,23]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one total parameter-independent statistic T=[0, 0, 22, 23] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 22, 23], [[0, [[0, 3/29], [22, 3/29], [23, 23/29]]], [1, [[0, 5/33], [22, 4/33], [23, 8/11]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 22, 23], [[0, [[0, 3/29], [22, 3/29], [23, 23/29]]], [1, [[0, 5/33], [22, 4/33], [23, 8/11]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0048","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cbd1d415f3a95153c893ead5bf3001a00cf5aafc85b9ef8945ccbb4c11a6477","family_id":"qf-mod36-sampling-family-readback","seed":202611056048,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,23,24]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one total parameter-independent statistic T=[0, 0, 23, 24] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 23, 24], [[0, [[0, 1/10], [23, 1/10], [24, 4/5]]], [1, [[0, 5/34], [23, 2/17], [24, 25/34]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 23, 24], [[0, [[0, 1/10], [23, 1/10], [24, 4/5]]], [1, [[0, 5/34], [23, 2/17], [24, 25/34]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0049","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3a5b46c41341cb4acc8da88bb1c3742f46d46f2ef6b85dfc4a091aea0df59c5","family_id":"qf-mod36-sampling-family-readback","seed":202611056049,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,24,25]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one total parameter-independent statistic T=[0, 0, 24, 25] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 24, 25], [[0, [[0, 3/31], [24, 3/31], [25, 25/31]]], [1, [[0, 1/7], [24, 4/35], [25, 26/35]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 24, 25], [[0, [[0, 3/31], [24, 3/31], [25, 25/31]]], [1, [[0, 1/7], [24, 4/35], [25, 26/35]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0050","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4006c9f9fbfa2b4a2c0b107c732ef95a74944778ee9a2f699ecf11138374c755","family_id":"qf-mod36-sampling-family-readback","seed":202611056050,"semantic_template":"qf-mod36-sampling-family-readback/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,25,26]}"},"fields":{"title":"One Fixed Statistic Has a Family of Parameter-Indexed Laws","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one total parameter-independent statistic T=[0, 0, 25, 26] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, parameter_laws, model_qualified, certificate).","answer":{"kind":"expression","value":"([0, 0, 25, 26], [[0, [[0, 3/32], [25, 3/32], [26, 13/16]]], [1, [[0, 5/36], [25, 1/9], [26, 3/4]]]], Yes, one_theta_independent_total_statistic_family_of_laws)","explanation":"Use the same statistic for every parameter row and report each pushforward in increasing parameter order. Do not average across parameters or add an unstated prior. The exact result, retaining the requested order, is ([0, 0, 25, 26], [[0, [[0, 3/32], [25, 3/32], [26, 13/16]]], [1, [[0, 5/36], [25, 1/9], [26, 3/4]]]], Yes, one_theta_independent_total_statistic_family_of_laws)."},"skills":["One Fixed Statistic Has a Family of Parameter-Indexed Laws"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0051","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baca53481c4c88d4f46e8c44e2cb294f13bd0527c31d2078d20b996494e2d738","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056051,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,8],[2,8],[3,8],[2,8]],[[2,12],[3,12],[4,12],[3,12]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,1,2]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 3/8], [1, 3/8], [2, 1/4]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 3/8], [1, 3/8], [2, 1/4]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0052","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f1428ee03a0240939983a8bb79cfb51f7334af4e2b4fea5bab36f91378fb5f","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056052,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,9],[2,9],[3,9],[3,9]],[[2,13],[3,13],[4,13],[4,13]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,2,3]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/9, 2/9, 3/9, 3/9], [2/13, 3/13, 4/13, 4/13]], and one total parameter-independent statistic T=[0, 0, 2, 3] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 5/13], [2, 4/13], [3, 4/13]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 5/13], [2, 4/13], [3, 4/13]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0053","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"508e1f5e3579f5a0921fc2651fea84d9e20ec3f423ad7397aa778aaae7439622","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056053,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,3,4]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one total parameter-independent statistic T=[0, 0, 3, 4] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 3, [[0, 3/10], [3, 3/10], [4, 2/5]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 3, [[0, 3/10], [3, 3/10], [4, 2/5]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0054","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7491645e7913eb0dcaf2f00505ead4092ba9e05b954eddd4cd853517ab37b17","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056054,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,4,5]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one total parameter-independent statistic T=[0, 0, 4, 5] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 5, [[0, 1/3], [4, 4/15], [5, 2/5]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 5, [[0, 1/3], [4, 4/15], [5, 2/5]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0055","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da8f9c97a9693ecf02c0f388876a891c694b242a837b21c88a33f6f6402b3287","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056055,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,5,6]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one total parameter-independent statistic T=[0, 0, 5, 6] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 1/4], [5, 1/4], [6, 1/2]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 1/4], [5, 1/4], [6, 1/2]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0056","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381fdd7207a4238f9fcbc9531a1c6a6a718c8ba301a1d08eaecde0e22825132b","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056056,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,6,7]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one total parameter-independent statistic T=[0, 0, 6, 7] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 5/17], [6, 4/17], [7, 8/17]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 5/17], [6, 4/17], [7, 8/17]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0057","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1006fcb283a0fe53f40ec1ce7a6bd657e1264e0dbd617d665154873d8b2d6696","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056057,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,7,8]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one total parameter-independent statistic T=[0, 0, 7, 8] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 7, [[0, 3/14], [7, 3/14], [8, 4/7]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 7, [[0, 3/14], [7, 3/14], [8, 4/7]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0058","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9481d516035ea71df721122b36df782cdae4a8d2a8dbcc7a6a70fb7db756a5ae","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056058,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,8,9]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one total parameter-independent statistic T=[0, 0, 8, 9] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 9, [[0, 5/19], [8, 4/19], [9, 10/19]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 9, [[0, 5/19], [8, 4/19], [9, 10/19]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0059","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aece709c295d73c058f922ef4bdb577a1c231e238a01106938c3183330d48bea","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056059,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,9,10]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one total parameter-independent statistic T=[0, 0, 9, 10] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 3/16], [9, 3/16], [10, 5/8]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 3/16], [9, 3/16], [10, 5/8]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0060","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b978cab1342ad3387e4484fd6ba6107a66f86e09ddbaa9eb1e9b7e75b99d6da1","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056060,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,10,11]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one total parameter-independent statistic T=[0, 0, 10, 11] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 5/21], [10, 4/21], [11, 4/7]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 5/21], [10, 4/21], [11, 4/7]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0061","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24e3ad7f334ad9b5c61822fb96645bc06589b608538ae3cb1e4f7445d906a68c","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056061,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,11,12]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one total parameter-independent statistic T=[0, 0, 11, 12] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 11, [[0, 1/6], [11, 1/6], [12, 2/3]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 11, [[0, 1/6], [11, 1/6], [12, 2/3]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0062","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f611da5077067e20a3b2027215cc4557aedee3426d31022122b8f1b2a338abf","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056062,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,12,13]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one total parameter-independent statistic T=[0, 0, 12, 13] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 13, [[0, 5/23], [12, 4/23], [13, 14/23]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 13, [[0, 5/23], [12, 4/23], [13, 14/23]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0063","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61bfa29e665f2f3f4c7578104019df65b1be65167ebfccc7af97493854cfd2fa","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056063,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,13,14]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one total parameter-independent statistic T=[0, 0, 13, 14] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 3/20], [13, 3/20], [14, 7/10]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 3/20], [13, 3/20], [14, 7/10]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0064","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8859393adaab5352e2c497aa321a21ec085bcffc97ad94a8c54e61fe500ae9ab","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056064,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,14,15]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one total parameter-independent statistic T=[0, 0, 14, 15] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 1/5], [14, 4/25], [15, 16/25]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 1/5], [14, 4/25], [15, 16/25]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0065","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a9a4711887b5e10a46627c3804c01782ef2cef4e4f6f7a7e2f691d6a95e8974","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056065,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,15,16]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one total parameter-independent statistic T=[0, 0, 15, 16] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 15, [[0, 3/22], [15, 3/22], [16, 8/11]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 15, [[0, 3/22], [15, 3/22], [16, 8/11]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0066","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9057fafb75bf49c82ba0d54b1f4f18d076397bd71bde15cfa2acebcb03542abb","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056066,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,16,17]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one total parameter-independent statistic T=[0, 0, 16, 17] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 17, [[0, 5/27], [16, 4/27], [17, 2/3]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 17, [[0, 5/27], [16, 4/27], [17, 2/3]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0067","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"695b70439e1eb502470ef2f9486e3c704bd744d70b57fdb68f0c6b813054feee","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056067,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,17,18]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one total parameter-independent statistic T=[0, 0, 17, 18] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 1/8], [17, 1/8], [18, 3/4]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 1/8], [17, 1/8], [18, 3/4]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0068","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e00418d5d1f0035d94870f75ecdd4c72e69a5bad357c0c23bc85e46af4c42e9b","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056068,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,18,19]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one total parameter-independent statistic T=[0, 0, 18, 19] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 5/29], [18, 4/29], [19, 20/29]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 5/29], [18, 4/29], [19, 20/29]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0069","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7993ea08ada979f4be2eb5b3c6b45800c79ebb0f34ca6dd887346027f91f02f","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056069,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,19,20]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one total parameter-independent statistic T=[0, 0, 19, 20] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 19, [[0, 3/26], [19, 3/26], [20, 10/13]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 19, [[0, 3/26], [19, 3/26], [20, 10/13]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0070","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"805604190c092ea39cdb730633c8d64b8c9efb1062bfd48664d705a94e8a65a3","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056070,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,20,21]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one total parameter-independent statistic T=[0, 0, 20, 21] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 21, [[0, 5/31], [20, 4/31], [21, 22/31]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 21, [[0, 5/31], [20, 4/31], [21, 22/31]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0071","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb80d0cff909580d3df91850f96521924f5384a3cbcb92c11a89f41c1bd98b4f","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056071,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,21,22]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one total parameter-independent statistic T=[0, 0, 21, 22] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 3/28], [21, 3/28], [22, 11/14]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 3/28], [21, 3/28], [22, 11/14]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0072","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a8e46f12253e310626309b5d73f6190838d662b74d59fcdaa94d42f49919632","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056072,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome\":1,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,22,23]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one total parameter-independent statistic T=[0, 0, 22, 23] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=1. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(1, 0, [[0, 5/33], [22, 4/33], [23, 8/11]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (1, 0, [[0, 5/33], [22, 4/33], [23, 8/11]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0073","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc92eceb160fdd690304becef7442c2868eb051005ce252ae5c90750f850fd0c","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056073,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome\":2,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,23,24]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one total parameter-independent statistic T=[0, 0, 23, 24] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=2. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(2, 23, [[0, 1/10], [23, 1/10], [24, 4/5]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (2, 23, [[0, 1/10], [23, 1/10], [24, 4/5]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0074","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f817cef3f033a4e7018341942f4c78ede08585f8e3ae9dd95c205032ba54d8fa","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056074,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome\":3,\"outcome_count\":4,\"parameter\":1,\"parameter_count\":2,\"statistic\":[0,0,24,25]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one total parameter-independent statistic T=[0, 0, 24, 25] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=1; supplied outcome index=3. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(3, 25, [[0, 1/7], [24, 4/35], [25, 26/35]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (3, 25, [[0, 1/7], [24, 4/35], [25, 26/35]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0075","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52b75df89abc2bdb87ecc8994f41b2090023ea05fd26ca41ceb2f4af9d680df5","family_id":"qf-mod36-sampling-observed-estimate-boundary","seed":202611056075,"semantic_template":"qf-mod36-sampling-observed-estimate-boundary/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome\":0,\"outcome_count\":4,\"parameter\":0,\"parameter_count\":2,\"statistic\":[0,0,25,26]}"},"fields":{"title":"An Observed Estimate Is Not the Whole Sampling Law","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one total parameter-independent statistic T=[0, 0, 25, 26] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Selected parameter=0; supplied outcome index=0. Give (outcome, estimate, law, boundary).","answer":{"kind":"expression","value":"(0, 0, [[0, 3/32], [25, 3/32], [26, 13/16]], observed_estimate_is_not_sampling_distribution)","explanation":"Read the statistic at the supplied outcome and retain the separate complete pushforward law at the selected parameter. One realized number does not replace that law or establish the unknown parameter. The exact result, retaining the requested order, is (0, 0, [[0, 3/32], [25, 3/32], [26, 13/16]], observed_estimate_is_not_sampling_distribution)."},"skills":["An Observed Estimate Is Not the Whole Sampling Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0076","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33deb547534b951dd9ddedea3b2cea8a92a579c4d1047d54878b08047e22049d","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056076,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,8],[2,8],[3,8],[2,8]],[[2,12],[3,12],[4,12],[3,12]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,1,2]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/8, 2/8, 3/8, 2/8], [2/12, 3/12, 4/12, 3/12]], and one total parameter-independent statistic T=[0, 0, 1, 2] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 1, 2], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 1, 2], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0077","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f37597fd0d3b07da79d685751c14bce8aa6f93997184682d172eb844cb3639","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056077,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,9],[2,9],[3,9],[3,9]],[[2,13],[3,13],[4,13],[4,13]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,2,3]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/9, 2/9, 3/9, 3/9], [2/13, 3/13, 4/13, 4/13]], and one total parameter-independent statistic T=[0, 0, 2, 3] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 2, 3], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 2, 3], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0078","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8410b3fd540c14979ccc37b139d62ecc7e79cf09575a9ff1ee4a418b8c5520af","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056078,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,3,4]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one total parameter-independent statistic T=[0, 0, 3, 4] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 3, 4], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 3, 4], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0079","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76a6c80ac50793807b011fdb647b1291980ccbce56197cbd734406bf291a033f","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056079,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,4,5]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one total parameter-independent statistic T=[0, 0, 4, 5] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 4, 5], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 4, 5], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0080","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d12493daed8107e575bc8c571e850b10a71ba269a0357ed34c11799b0a63f6b","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056080,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,5,6]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one total parameter-independent statistic T=[0, 0, 5, 6] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 5, 6], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 5, 6], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0081","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"884dff0b6344868de758b33eb35afa751b97df34d5ac859a04e83508f9cc0e7a","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056081,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,6,7]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one total parameter-independent statistic T=[0, 0, 6, 7] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 6, 7], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 6, 7], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0082","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3befc5748d3388aa99c49daab43bf4ae7271299d71a0f9f34813c597712e41","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056082,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,7,8]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one total parameter-independent statistic T=[0, 0, 7, 8] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 7, 8], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 7, 8], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0083","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da7fd57d9b81aa5f75640e39c5c4ea4a05ee971424946b109a2115201aa6267d","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056083,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,8,9]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one total parameter-independent statistic T=[0, 0, 8, 9] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 8, 9], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 8, 9], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0084","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3cfa421764f7d5f117fbbe75b2e66bf6681bdaf210eae3cc3ea29ae3c5c9d58","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056084,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,9,10]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one total parameter-independent statistic T=[0, 0, 9, 10] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 9, 10], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 9, 10], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0085","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8702404ad16c612ad26528885d730a67a0f01422698c06b4de58dcef489736c7","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056085,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,10,11]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one total parameter-independent statistic T=[0, 0, 10, 11] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 10, 11], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 10, 11], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0086","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2542a294b7e4a516778f7569c165d9035303491caa3540a9c41cfab7e25a585","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056086,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,11,12]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one total parameter-independent statistic T=[0, 0, 11, 12] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 11, 12], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 11, 12], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0087","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b16190443ff2892fe524546b65b946d10d1b9df18bd4b02d65716fe1f4b3c728","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056087,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,12,13]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one total parameter-independent statistic T=[0, 0, 12, 13] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 12, 13], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 12, 13], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0088","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"940e117ffc1212b01544c386355144786836cb5e78a3aa1cc2147f33f70a1a31","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056088,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,13,14]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one total parameter-independent statistic T=[0, 0, 13, 14] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 13, 14], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 13, 14], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0089","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d12a754bfcd4cf5211b3a58c4dbce38e955450dd4fee1dc9e1e02fe6e1e6457","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056089,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,14,15]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one total parameter-independent statistic T=[0, 0, 14, 15] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 14, 15], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 14, 15], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0090","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b6b6d33353d69dfa0dbebd01840387c5ce25c11f6b3d17b59368111cef780c0","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056090,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,15,16]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one total parameter-independent statistic T=[0, 0, 15, 16] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 15, 16], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 15, 16], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0091","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bed8b445ee60abae9be5c92f3a0294eaa82e26dd655599c725c03b7056f52e4","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056091,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,16,17]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one total parameter-independent statistic T=[0, 0, 16, 17] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 16, 17], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 16, 17], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0092","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"416a953c33fd295b8b0b57c54dc3ea56a79551b3359e04f829ec9739999590ba","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056092,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,17,18]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one total parameter-independent statistic T=[0, 0, 17, 18] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 17, 18], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 17, 18], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0093","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b5b7017633f4b434bcf24743b197118f0f84dec4b90564801ea8e6b08865e60","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056093,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,18,19]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one total parameter-independent statistic T=[0, 0, 18, 19] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 18, 19], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 18, 19], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0094","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aadfc2d448e99cd24a8bd6d9134dc5cb16bdf00f52d513a47ba61a4d01dffdd","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056094,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,19,20]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one total parameter-independent statistic T=[0, 0, 19, 20] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 19, 20], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 19, 20], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0095","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b8f7872188af43677672a1f823fcc29a5def0af6dc27690b281eb311c315f56","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056095,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,20,21]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one total parameter-independent statistic T=[0, 0, 20, 21] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 20, 21], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 20, 21], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0096","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5da3b332dc16c9563f7339994e5220693357bbd407426934b36c1ba5dfd1c80","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056096,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,21,22]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one total parameter-independent statistic T=[0, 0, 21, 22] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 21, 22], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 21, 22], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0097","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"399aeec189b37d9367ecc58d4ab57a1e04d11e9695998a1c68aa64c772539045","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056097,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,22,23]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one total parameter-independent statistic T=[0, 0, 22, 23] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 22, 23], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 22, 23], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0098","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eaf8d7066cc56e892e03a1a4d2cc93a25d80d213b278087757e8dc09c04552a","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056098,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,23,24]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one total parameter-independent statistic T=[0, 0, 23, 24] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 23, 24], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 23, 24], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0099","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c733b10fb5b08a2b52902e483c50452bd7bdf3a7841de9c936d871b0b12d1971","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056099,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,24,25]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one total parameter-independent statistic T=[0, 0, 24, 25] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 24, 25], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 24, 25], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r06-sampling-distribution-0100","batch_id":"batch-20261014-advance-20261105-r06-sampling-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa7a567b738620714e704a2c053e42b73d3df074a32ad9e0d501f9247ab6df86","family_id":"qf-mod36-sampling-property-nonconclusion","seed":202611056100,"semantic_template":"qf-mod36-sampling-property-nonconclusion/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome_count\":4,\"parameter_count\":2,\"statistic\":[0,0,25,26]}"},"fields":{"title":"Withholding Statistical Claims Does Not Establish Their Negations","stem":"Use common outcomes=Fin(4)={0,…,3}, parameter indices=Fin(2)={0,…,1}, complete exact law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one total parameter-independent statistic T=[0, 0, 25, 26] in outcome order. Every law is nonnegative and sums to1. For each parameter, add law masses over each T-fibre; list the distinct statistic values increasingly, retaining zero-mass fibres. A parameter law, a sampling law and an observed estimate are different objects; no prior or posterior is supplied. Give (statistic, claim_marker_1, claim_marker_2, claim_marker_3, claim_marker_4, boundary).","answer":{"kind":"expression","value":"([0, 0, 25, 26], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)","explanation":"Retain the four literal No nonclaim markers. The source boundary lists absent unbiasedness, low-variance, consistency, likelihood and posterior claims; it does not supply four named tests or prove any of those properties false. The exact result, retaining the requested order, is ([0, 0, 25, 26], No, No, No, No, no_unbiased_low_variance_consistency_likelihood_or_posterior_claim)."},"skills":["Withholding Statistical Claims Does Not Establish Their Negations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}