{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r03-estimator-bias-0001","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df49f2ae6c1fefb28444221f3f78ff6982b17a9f626354913ce21f4304f1f0d7","family_id":"qf-mod37-bias-at-parameter","seed":202611063001,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"parameter\":0,\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0002","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb10acf7dc01026dd335e8e12bfdc71f87ac430d6894db960fa9571c8ae23811","family_id":"qf-mod37-bias-at-parameter","seed":202611063002,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"parameter\":1,\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0003","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bd65a66f479b12077debd6e38d15b16c4bbe07878cde7d00bc8918b7984c8d6","family_id":"qf-mod37-bias-at-parameter","seed":202611063003,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"parameter\":0,\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0004","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76c8a91e20edcd6325eb69fbbf40e53bde03dd60620a81fea442a67dcb0be527","family_id":"qf-mod37-bias-at-parameter","seed":202611063004,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"parameter\":1,\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0005","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fe75f49a7df00ff19276c35cf8301488008b2ed26b87333218bb4e171c84938","family_id":"qf-mod37-bias-at-parameter","seed":202611063005,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"parameter\":0,\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0006","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44c0cf765e4c604a0ad2ccae1a69c87df16cfce85d3694cfe9cc6623353e5d03","family_id":"qf-mod37-bias-at-parameter","seed":202611063006,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"parameter\":1,\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0007","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb1d70a87f1b4ded4da5dbd79a6f13b6875ebe5e2c82ac37145f8b3edaeddb3","family_id":"qf-mod37-bias-at-parameter","seed":202611063007,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"parameter\":0,\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0008","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9f76635accdd5487a898aad1aa6ef5898ea4623e184a478d3a9f1b47253128e","family_id":"qf-mod37-bias-at-parameter","seed":202611063008,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"parameter\":1,\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0009","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"750aa5f479c2dba7749cd9e3af7ac71766488b366858c529c767884e41a9bd26","family_id":"qf-mod37-bias-at-parameter","seed":202611063009,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"parameter\":0,\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0010","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d927f2d2ce155ed3746683f975ef13165106aeb3895338e36f1238d96fc03e8c","family_id":"qf-mod37-bias-at-parameter","seed":202611063010,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"parameter\":1,\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0011","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12cf51a36a38cce380a35fa64c027cf8a70f6f24a4c27b107b430e653bd5fb00","family_id":"qf-mod37-bias-at-parameter","seed":202611063011,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"parameter\":0,\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0012","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdfda9fc7a7eba0f228e9620a24b77e28bd6f678848dc9bb0223fd723fb6a663","family_id":"qf-mod37-bias-at-parameter","seed":202611063012,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"parameter\":1,\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0013","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e38df216427819d8b8f3b632b5bc5a1e943c70b39a7b8f0219c19026f533c601","family_id":"qf-mod37-bias-at-parameter","seed":202611063013,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"parameter\":0,\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0014","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa5caf7cd94a4ec602c7846c3ad7a0f9b5afd1f60dd67e9eb6e3bd45ff4e2a56","family_id":"qf-mod37-bias-at-parameter","seed":202611063014,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"parameter\":1,\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0015","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69c8423a55f0275538a6c07b0229a17f22c5068bf66c85693a83c00a3f1e196f","family_id":"qf-mod37-bias-at-parameter","seed":202611063015,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"parameter\":0,\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0016","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"816dff555e3a6f85167b437baa7a314e2aceaf4e9a4a6af45a5ce41b4b9974bb","family_id":"qf-mod37-bias-at-parameter","seed":202611063016,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"parameter\":1,\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0017","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"391b9c128e1ee32c3981972358dc94f5967e03b49f4840d50f6f57a5ae9238cd","family_id":"qf-mod37-bias-at-parameter","seed":202611063017,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"parameter\":0,\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0018","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4e1b14b4980792e035bb36985e49488694354718b6783f9bb4531fe64ae707d","family_id":"qf-mod37-bias-at-parameter","seed":202611063018,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"parameter\":1,\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0019","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"547ef4fe85c095fb0c7a40c0d9b7819585bf39f327874658efb4bf2f22f65db1","family_id":"qf-mod37-bias-at-parameter","seed":202611063019,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"parameter\":0,\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0020","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e03a6decd8df64baf2170adbd6c01ec5a66bb605eea5cf56a8d64e1607f5fe24","family_id":"qf-mod37-bias-at-parameter","seed":202611063020,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"parameter\":1,\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0021","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b2e26a8891b07dc0e5c2696d17793321ff2cf5408cd8836d4d86655b62d7c4","family_id":"qf-mod37-bias-at-parameter","seed":202611063021,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"parameter\":0,\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0022","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce73005bb7de14ce0c9879e387e137283498e20082c1be6449f89ba3c70278c","family_id":"qf-mod37-bias-at-parameter","seed":202611063022,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"parameter\":1,\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0023","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"503f1b53deb76082b9d677bb7a857ddef49d2e9ecfe90cf7a0175dc40cd387b7","family_id":"qf-mod37-bias-at-parameter","seed":202611063023,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"parameter\":0,\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0024","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e4b7b9b1895063d139e8dbc9465c2672a94c472625409666caac6b0f15d42e","family_id":"qf-mod37-bias-at-parameter","seed":202611063024,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"parameter\":1,\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0025","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce94b47f1d636e7e8efea8041b7a219adccca0abeef75962e5361a321ba9721c","family_id":"qf-mod37-bias-at-parameter","seed":202611063025,"semantic_template":"qf-mod37-bias-at-parameter/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"parameter\":0,\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"},"fields":{"title":"Subtract the Target from the Expectation under That Same Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)","explanation":"Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)."},"skills":["Subtract the Target from the Expectation under That Same Parameter"],"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-20261106-r03-estimator-bias-0026","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac3f9609021e42039c6918f57e7bdcab92198fb756e8d60903d20c4333e31968","family_id":"qf-mod37-bias-family-readback","seed":202611063026,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0027","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d33d256830e4c7404e64dc0edae04c6859bd5b61b7a69ff53bb0323c779efcbb","family_id":"qf-mod37-bias-family-readback","seed":202611063027,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0028","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21eacdfd8ef6e347f59a01ce831a8ade66122e1e7b1ed0c8f1d41e75b2ae6323","family_id":"qf-mod37-bias-family-readback","seed":202611063028,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0029","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9bef6ee46d73a94156d519d2080213d03f8e7a8842c2b3227c08b1ed29375a0","family_id":"qf-mod37-bias-family-readback","seed":202611063029,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0030","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09c0761d622ca77b8e9b5f9a143225dc3121dbb45341b1710fe853fd0c4d7a8a","family_id":"qf-mod37-bias-family-readback","seed":202611063030,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0031","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aae9d3b62d421fe53a652b030b144c30f0911cef3623cef800c77a7fceb6ab9","family_id":"qf-mod37-bias-family-readback","seed":202611063031,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0032","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8220c1bf20438f061d9be45adb83cb733dc7ac20829bdc286d8113f61821db67","family_id":"qf-mod37-bias-family-readback","seed":202611063032,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0033","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf73ee917116d0f129987c5c5df3fec86510a01879ee17c4b8fe5d0f84de9ceb","family_id":"qf-mod37-bias-family-readback","seed":202611063033,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0034","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96a5e1342a20db21434e1f452c16ffee6f7c89cc7f0820954a67792ad42548a6","family_id":"qf-mod37-bias-family-readback","seed":202611063034,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0035","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e79f97919260811698bc5e251dde304646e77e604909e6ad734738ecdc3fc882","family_id":"qf-mod37-bias-family-readback","seed":202611063035,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0036","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57c930d8fec8ea0af5db52ab9bfeb47cdc1904cfc2c2080e11b96cca75321235","family_id":"qf-mod37-bias-family-readback","seed":202611063036,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0037","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70e3d6895be61e014e2efca9178663876856f4366849b7e67765e4e25d69152d","family_id":"qf-mod37-bias-family-readback","seed":202611063037,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0038","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f0b708ca91f837ea9aec300d766584f3ee60891114c973b70743bfb65a2f237","family_id":"qf-mod37-bias-family-readback","seed":202611063038,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0039","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65721ea3b0b38d505c71815936c3ff5383858bc04ca4d944ad6fb79d5a665958","family_id":"qf-mod37-bias-family-readback","seed":202611063039,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0040","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd0a822d723f0963ce7c023104faa713f9759de2837307791b959069d9733b43","family_id":"qf-mod37-bias-family-readback","seed":202611063040,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0041","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6619e0859887f49c7f4722b4cf38aef1550bcbf5622e0a090d350ae2c99fd3e","family_id":"qf-mod37-bias-family-readback","seed":202611063041,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0042","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c49e3f866aa487b0ee94b1c2ca1d5ab8ceb30786291297675e4ab0f053c8f68","family_id":"qf-mod37-bias-family-readback","seed":202611063042,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0043","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"969b165fc09186f7bb2aab55c122ab5e0a4cbdcd7b31e81ae7362ba2559de61c","family_id":"qf-mod37-bias-family-readback","seed":202611063043,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0044","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada23100673e09a53be35d9cdfded2c6f1d2e89ffefa9f4fb6075ed8b3cc3049","family_id":"qf-mod37-bias-family-readback","seed":202611063044,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0045","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e854d40a86b27413034dbb56817ee5c407b02fe58b32ea438859adaf05b8a5a9","family_id":"qf-mod37-bias-family-readback","seed":202611063045,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0046","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27dbd7e9c5665094d32db58a40e1d85c0e365628526014e36e0e27f5b4b774b2","family_id":"qf-mod37-bias-family-readback","seed":202611063046,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0047","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f38fbe22ad6174a45765c16c0e5bea2c11781a2f9d88399cb201760b286eb6a","family_id":"qf-mod37-bias-family-readback","seed":202611063047,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0048","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c15c95ead0901845cda446649e05e0c94f80c9bffcb4d86b0ea39cd4b0505bc","family_id":"qf-mod37-bias-family-readback","seed":202611063048,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0049","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"352f3adc34cef1336b2d751fb11cc3e2f03988d5f627b8bdaf80625afd2477d1","family_id":"qf-mod37-bias-family-readback","seed":202611063049,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0050","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0871aae01e65d1a7240415b6115350f2d335e1000fa83cd85811d157a5e90fba","family_id":"qf-mod37-bias-family-readback","seed":202611063050,"semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"},"fields":{"title":"Bias Is an Indexed Family, Not One Observed Error","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).","answer":{"kind":"expression","value":"([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)","explanation":"For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)."},"skills":["Bias Is an Indexed Family, Not One Observed Error"],"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-20261106-r03-estimator-bias-0051","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8972bb337afd20b59976318f9ce202df61782cadabe3b7834f3377386cbf1615","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063051,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0052","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544f57923263fb5b29f572521b47ac90299bfda2c6fbde16a6dd8215cfe09e8a","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063052,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0053","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"666008d39d782b2f0eaf076bc9e990d588681e93dd5fe66860416892d8d1e806","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063053,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0054","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72ed18493785a0f3c8e45e35f2d605ddfe846f4f93fe9a5246dfcc585d3fd61","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063054,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0055","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbe578b760d5ce0a1fac15637a728c8fdeafa3d48b802381d6833ec4bff8b446","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063055,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0056","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4957e7eba53447b6626fe771c90b154ba5a5a0ae0e7a40b7f8e86fcaa74cd25e","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063056,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0057","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef5405a991c30504b7fb100832cee80faa8bb504a95f2c12e56624854900579d","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063057,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0058","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"085a9409cc720cef949f8282b4dd32df332163b6261c277dc31c6169c30477e4","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063058,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0059","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a5d6f92f2d9895128f7075e3f211a8aef20821c6c8c007a16a08852706077b6","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063059,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0060","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1055c3d2aa4b348edfcc320e62abd5503cbc48f81aea33463c70b0359140826d","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063060,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0061","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ff4762cc30f7efd398596d85497f5b49a807a7bad3df5777fec31054664082d","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063061,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0062","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4715b7732d456a086dbfa6360e76fd3df4d111d343ae600da2ac5c16b50c7261","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063062,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0063","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd802fd767d5703ad14bd488e61d965bc179cd63ed5179ab113a19ecaf4a50aa","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063063,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0064","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54e2edba253bc7e6f41c90aa16b4d0d2e6a2eccf341683275b773c7d080514b6","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063064,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0065","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c8f92d9ddfb387d3d9d529767ee29c63afb70662554e5e52459297005dca565","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063065,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0066","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e61d2e8432754554f9e9acc6804f3d30c7d0d8dfa79ddda1576bc664b8504a67","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063066,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0067","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f826b129931055c84c64868d588baa9f57a6e5f1764b884de09ce314096e2bd","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063067,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0068","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8edf8ee18ff264d60841af7781c60f055849f17c1689eeb6c7c4822672b139a7","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063068,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0069","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b12af5e11a511cff0df2c3a15cfcc4e0aa27a00d7cdc9cf845ccc740dbec2a","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063069,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0070","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d962373d3ae4ea0c737fda4937d7b1a635e0a2b98dcede99fb53436f5f49d9a8","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063070,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0071","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a234ebeed67715d42b643b10031a95f1bcfb40173ed11f85dd125025bff86e82","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063071,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0072","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a0555cb547febcccf140b90cd3475287bd017a8b95b932d04af15376257cb2b","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063072,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0073","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23e123c0ee70c52126d467808539db6d411fb42cdf409f4d39150c1256737ddc","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063073,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0074","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ea9caa255fb12c2d17f97df3a5ff6ed4833215546a19fb99120e81d9cfaa39","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063074,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0075","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c9a57b77a8fe0909edd86e692524da98db0fecb2bebea6f48fd75314eeb837e","family_id":"qf-mod37-bias-unbiased-audit","seed":202611063075,"semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"},"fields":{"title":"Unbiasedness Requires Zero Bias at Every Declared Parameter","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).","answer":{"kind":"expression","value":"([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta)","explanation":"Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta)."},"skills":["Unbiasedness Requires Zero Bias at Every Declared Parameter"],"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-20261106-r03-estimator-bias-0076","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7cbd5f98a44cdbcc3f9c4b47459082fd9799a74b0cdd1045c549e1d23fc242a","family_id":"qf-mod37-bias-observed-boundary","seed":202611063076,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0077","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f92f49ef367f1b6edee3217c7bc8a4f4b87c3347dfbfb5e846a41d55bb0d1da7","family_id":"qf-mod37-bias-observed-boundary","seed":202611063077,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0078","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d2cff8b559f498a330cb3a5e54e33f286d98c25a5d6558524373a1d30ab2ac","family_id":"qf-mod37-bias-observed-boundary","seed":202611063078,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0079","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11fefea48452d8b96e5395ae95f338e462bdbb1a7e615503f82bda435ff2c7c3","family_id":"qf-mod37-bias-observed-boundary","seed":202611063079,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0080","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dbe14d4479611e6fba80509326bef164b792c1ce10c99928b09f9caa5269300","family_id":"qf-mod37-bias-observed-boundary","seed":202611063080,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0081","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9790ac4706f841b9f4e3f995075282cf5fc6f9472fa0d3a0393e20eb84a8cc39","family_id":"qf-mod37-bias-observed-boundary","seed":202611063081,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0082","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40b97466165d32887cddc5113b0d9c4cb449d4139177c0e8b02c2d86ba46e543","family_id":"qf-mod37-bias-observed-boundary","seed":202611063082,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0083","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5adbcfba9557584ffcc087f602925abd501106ea8c338850f2bb7cf368946b9b","family_id":"qf-mod37-bias-observed-boundary","seed":202611063083,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0084","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1787cba04cdd743a49b8d38455c8b0784f0e2569ac22d244f78ff1db714304e1","family_id":"qf-mod37-bias-observed-boundary","seed":202611063084,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0085","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1d0442d8f0ee9e5126e9d509d41a785ff3d9a749a962a671cb702c51a66393a","family_id":"qf-mod37-bias-observed-boundary","seed":202611063085,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0086","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c48807f3beefc6585cd24711d305dd40f8a06910b64f061db7465945f77cd3dc","family_id":"qf-mod37-bias-observed-boundary","seed":202611063086,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0087","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04e58ddf4536d53f74b1ed878434c9ffdc1ed0ff1982c09679216baf85de6ad3","family_id":"qf-mod37-bias-observed-boundary","seed":202611063087,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0088","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51931dcc800f5f6aac2ad77eef546a1b8d6a4cd58db7166f5f857ae7813c7288","family_id":"qf-mod37-bias-observed-boundary","seed":202611063088,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0089","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b663b4cc249dc669e287138ea86f09ad910258902fabb72db9c6ee815b889ed7","family_id":"qf-mod37-bias-observed-boundary","seed":202611063089,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0090","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de35daf41d5506625ef881109214c7d79584b7ab216d57ed98ee05a35da64479","family_id":"qf-mod37-bias-observed-boundary","seed":202611063090,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0091","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"256de748b6d1c4693f7092e7c57ab5bb63d401ee53262d642c7b9917db92769c","family_id":"qf-mod37-bias-observed-boundary","seed":202611063091,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0092","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2c838fa665aa5837ce45d18c96c94cde70b87482313696383832696fd29f741","family_id":"qf-mod37-bias-observed-boundary","seed":202611063092,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0093","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514749a2c6abc411ed48db55749020ff1e2c5f266b82abd4586bfa5d93e0ebfe","family_id":"qf-mod37-bias-observed-boundary","seed":202611063093,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0094","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3b94767c113418d3725fed0cbbe02cabea4e527f8d01aaf3619158f694ba59","family_id":"qf-mod37-bias-observed-boundary","seed":202611063094,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0095","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92f16521ebd9a74768ab35deddd6e65ffc3056fa374a0a983b50f9309f9b961b","family_id":"qf-mod37-bias-observed-boundary","seed":202611063095,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0096","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"395bae7f22040810d3d3adac5427e611386fe2cc6647fec8ec32ea8da366a060","family_id":"qf-mod37-bias-observed-boundary","seed":202611063096,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0097","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3039a0fe4962c5e1a6cc2dcaa7d06bfa66110224735487a5e28a28579df2c4fe","family_id":"qf-mod37-bias-observed-boundary","seed":202611063097,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0098","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0184de19ca2cead94946aabd389da85c52ab7ea318645c734b2cbbe9e9f970c2","family_id":"qf-mod37-bias-observed-boundary","seed":202611063098,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0099","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72d5b33983f6897541931009c0bc59896be32017298261563ca9dd592c91b117","family_id":"qf-mod37-bias-observed-boundary","seed":202611063099,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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-20261106-r03-estimator-bias-0100","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31ebb3cbd9cfb5613d7e87d9342ba347c9e8a56976f6700e21b1db8df9c72ca5","family_id":"qf-mod37-bias-observed-boundary","seed":202611063100,"semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"},"fields":{"title":"One Realized Estimate Does Not Establish Bias or Other Properties","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).","answer":{"kind":"expression","value":"(0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation":"Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)."},"skills":["One Realized Estimate Does Not Establish Bias or Other Properties"],"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."}}