{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0001","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb78d88a36629fbd1aa6e649f054aa1808a8f4b9908172e8d9de241e1dda733f","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183001,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,3],[2,3]],[[2,3],[1,3]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[1/3, 2/3], [2/3, 1/3]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[1/3, 2/3], [2/3, 1/3]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0002","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"085224b71e3ae6f0c2224114cedb76b123dc620145a0b19bad2df01c911cf0a0","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183002,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,2],[1,2]],[[1,2],[1,2]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0003","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15765bfec0c82ca8357ed311d150d6ea4db0c9ef5dc599d75db9a2928d8ccf07","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183003,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,5],[2,5]],[[2,5],[3,5]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[3/5, 2/5], [2/5, 3/5]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[3/5, 2/5], [2/5, 3/5]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0004","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"826c2b89b7ec0ee5cda15a08382b72d0d43b8ac7d3d7be668a048e66933b7591","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183004,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[2,3],[1,3]],[[1,3],[2,3]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[2/3, 1/3], [1/3, 2/3]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[2/3, 1/3], [1/3, 2/3]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0005","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1a01dfd073f3067427ca7b519188f2225830cfcd8ac0db7a33ee5948d9b9bb6","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183005,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,7],[2,7]],[[2,7],[5,7]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[5/7, 2/7], [2/7, 5/7]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[5/7, 2/7], [2/7, 5/7]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0006","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f43fd8ab58019786918b1b45e7beac845428a2528acd50b650e7cbc574da43a","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183006,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,4],[1,4]],[[1,4],[3,4]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0007","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34deda4098715d3679eb3fbce37bada166cb660a3d6701317d2da2ef95e14044","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183007,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,9],[2,9]],[[2,9],[7,9]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[7/9, 2/9], [2/9, 7/9]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[7/9, 2/9], [2/9, 7/9]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0008","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a86274be19fb6b2fcf82813a81235e0dabb73450b5f9339c7f7509fdddea100c","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183008,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[4,5],[1,5]],[[1,5],[4,5]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[4/5, 1/5], [1/5, 4/5]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[4/5, 1/5], [1/5, 4/5]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0009","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c82806b1e0f382203aa2bf043e5eba2f21caf159d544f35f37db0d80c9cf0b3","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183009,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,11],[2,11]],[[2,11],[9,11]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[9/11, 2/11], [2/11, 9/11]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[9/11, 2/11], [2/11, 9/11]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0010","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c8cb93608ea1496f88e62a16b325e973e07db18cfeba01ea2a4807d298679e","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183010,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,6],[1,6]],[[1,6],[5,6]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[5/6, 1/6], [1/6, 5/6]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[5/6, 1/6], [1/6, 5/6]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0011","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9b932548257d1bfdf78b58686fe3782ca714bf814ca433b52a94e2f49053418","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183011,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,13],[2,13]],[[2,13],[11,13]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0012","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aa4e195502c7f5f09193f8eda181a15bced3b83939016b1232f8f5bfa2084bf","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183012,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[6/7, 1/7], [1/7, 6/7]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[6/7, 1/7], [1/7, 6/7]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0013","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2283518f20cc0b9fce6bcc76fdf99a063b0c3768692b1f88a4bfb26d1e0da97","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183013,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[13,15],[2,15]],[[2,15],[13,15]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[13/15, 2/15], [2/15, 13/15]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[13/15, 2/15], [2/15, 13/15]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0014","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11c4e3b0922dc1246b7887457f96d35368a61760ea10510080562239033f9cd3","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183014,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0015","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c05113aa808fcc2814b372c916c8d66d2ce795c6f2d1da98f5422b3b1e038ce9","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183015,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[15,17],[2,17]],[[2,17],[15,17]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[15/17, 2/17], [2/17, 15/17]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[15/17, 2/17], [2/17, 15/17]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0016","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61bec1cb03332a90d7e95a01d7613d3f9b5458b3a709e37e04436c10c0b38549","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183016,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[8,9],[1,9]],[[1,9],[8,9]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[8/9, 1/9], [1/9, 8/9]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[8/9, 1/9], [1/9, 8/9]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0017","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07994e9053471f595a58dd3ba14c3cf626f02eda0fee6f2b81102f78c9eee30e","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183017,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[17,19],[2,19]],[[2,19],[17,19]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[17/19, 2/19], [2/19, 17/19]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[17/19, 2/19], [2/19, 17/19]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0018","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10e0e062659efe28636ccec9ad62f495dcbc36b33dda0f2863f819b885ab54e6","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183018,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,10],[1,10]],[[1,10],[9,10]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[9/10, 1/10], [1/10, 9/10]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[9/10, 1/10], [1/10, 9/10]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0019","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"365c3cf1d49c512513b76c140148f04319a1a9196a90f4e5d0095eb6815aad6e","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183019,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[19,21],[2,21]],[[2,21],[19,21]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[19/21, 2/21], [2/21, 19/21]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[19/21, 2/21], [2/21, 19/21]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0020","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6cb979c663a159b8b11cda07ebb06832c5b2f68906af783bcbfde80a522f4cf","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183020,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[10,11],[1,11]],[[1,11],[10,11]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[10/11, 1/11], [1/11, 10/11]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[10/11, 1/11], [1/11, 10/11]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0021","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3879c1d612194b6c568fda4df88f1d3335fea0f98f873ce782ab7a94f2ed312","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183021,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[21,23],[2,23]],[[2,23],[21,23]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[21/23, 2/23], [2/23, 21/23]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[21/23, 2/23], [2/23, 21/23]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0022","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83260d0e2cdf40fbc5767d5165d814e3a20ad7f7ccfd39ba8d0fe92a2dfc1c5f","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183022,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,12],[1,12]],[[1,12],[11,12]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[11/12, 1/12], [1/12, 11/12]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[11/12, 1/12], [1/12, 11/12]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0023","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14652306652e9eb3fc956972fa473dd3bbb348a99c62c5cdfae0327109d71821","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183023,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[23,25],[2,25]],[[2,25],[23,25]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[23/25, 2/25], [2/25, 23/25]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[23/25, 2/25], [2/25, 23/25]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0024","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544bba0ed88b3eb5939159e1297a9d24e6fec05cc7632162a23f7f2a5ca25aed","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183024,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[12,13],[1,13]],[[1,13],[12,13]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[12/13, 1/13], [1/13, 12/13]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[12/13, 1/13], [1/13, 12/13]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0025","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b8f2bb024bd0397fab84936e61f6564aa96a7274f8ff6ff72c0cacb10f6f9cf","family_id":"qf-mod49-ci-procedure-qualification","seed":202611183025,"semantic_template":"ci-procedure-qualification|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[25,27],[2,27]],[[2,27],[25,27]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"A Complete Precommitted Interval Procedure Keeps Every Probability Alignment","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).","answer":{"kind":"expression","value":"([0, 1], [a, b], [[0, 0], [1, 1]], [[25/27, 2/27], [2/27, 25/27]], Yes, total_precommitted_theta_independent_interval_procedure)","explanation":"Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[25/27, 2/27], [2/27, 25/27]], Yes, total_precommitted_theta_independent_interval_procedure)."},"skills":["A Complete Precommitted Interval Procedure Keeps Every Probability Alignment"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0026","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eab39f9b3b491b8a7a5c93e0cd2877d494816edc7292e146fdb4da8c469a5cf","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183026,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,3],[2,3]],[[2,3],[1,3]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 1/3], [1, 1/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 1/3], [1, 1/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0027","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d18dbf75c967510d4887d6d5fbf0f74afe98bad5e253e633da1593580463a0b6","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183027,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,2],[1,2]],[[1,2],[1,2]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 1/2], [1, 1/2]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0028","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2f3fdccb61d90de096c6e2fe2e03b068e1c378113d26d0c87c909ef7f722118","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183028,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,5],[2,5]],[[2,5],[3,5]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 3/5], [1, 3/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 3/5], [1, 3/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0029","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff1208e1f9a49389f7539a667d7b5188d6b9e90cd2c33f082053d02e78ea6faf","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183029,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[2,3],[1,3]],[[1,3],[2,3]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 2/3], [1, 2/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 2/3], [1, 2/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0030","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8aaceae32c5eee9aed55c2be067c369195b9e56c733e3fa0393042f7efcce77","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183030,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,7],[2,7]],[[2,7],[5,7]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 5/7], [1, 5/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 5/7], [1, 5/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0031","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a5e31cec370653da7d11e1b8cc82ff91ed898788c5954c95b263dc5d0166886","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183031,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,4],[1,4]],[[1,4],[3,4]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 3/4], [1, 3/4]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 3/4], [1, 3/4]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0032","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408e8c61987161fbc0374814b5957d532827108971f9bd03b8c60ed911266f27","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183032,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,9],[2,9]],[[2,9],[7,9]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 7/9], [1, 7/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 7/9], [1, 7/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0033","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38fef88ba3416c16d7e8e94bb4430cd2f0a2402b78456238b8a76147336350c0","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183033,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[4,5],[1,5]],[[1,5],[4,5]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 4/5], [1, 4/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 4/5], [1, 4/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0034","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0a9222c00b5e50ec5f3f58b76c24f985e1581659fb0f2d703f45e90473efd41","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183034,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,11],[2,11]],[[2,11],[9,11]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 9/11], [1, 9/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 9/11], [1, 9/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0035","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9704e9acb0d88fa94c10b178caa4868548a8e893eb018ec42bdf0654a77264eb","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183035,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,6],[1,6]],[[1,6],[5,6]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 5/6], [1, 5/6]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 5/6], [1, 5/6]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0036","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0a7faff967ba094bafd5b76ec8fd788e192dd310abce55079e8faf20c7a27b4","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183036,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,13],[2,13]],[[2,13],[11,13]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 11/13], [1, 11/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 11/13], [1, 11/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0037","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd407dd13d7d2a71a5b5bfac36497baca4c10bcc24dada2209354a5dab128e5","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183037,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 6/7], [1, 6/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 6/7], [1, 6/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0038","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb61d168e18e904135745f1d034069d594460e6761fc3cdebdea18f29f308128","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183038,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[13,15],[2,15]],[[2,15],[13,15]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 13/15], [1, 13/15]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 13/15], [1, 13/15]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0039","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d180c6b618f1b6fb914a86156764e200c0f1cb019d41e9bfcc266038ac2dabeb","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183039,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 7/8], [1, 7/8]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 7/8], [1, 7/8]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0040","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7350eac659f3cd076d4c3e7de22e22a0aa195dafabae084cb62984cfccb3a906","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183040,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[15,17],[2,17]],[[2,17],[15,17]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 15/17], [1, 15/17]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 15/17], [1, 15/17]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0041","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a23866dfd0822aa8474f49d0af4c0bc1530e1039ffe53733de6bc21d864738ee","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183041,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[8,9],[1,9]],[[1,9],[8,9]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 8/9], [1, 8/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 8/9], [1, 8/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0042","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51f1057c262528f1dc521c30946fb0504c0852ddceffd141db079ff84e93f6e0","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183042,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[17,19],[2,19]],[[2,19],[17,19]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 17/19], [1, 17/19]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 17/19], [1, 17/19]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0043","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"887e28b141ac918e53160dbb5fc51c116d17c475742c2e6b3e15a2ca9f22ab34","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183043,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,10],[1,10]],[[1,10],[9,10]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 9/10], [1, 9/10]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 9/10], [1, 9/10]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0044","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc6d358260a748b91a8038632a24c77299cffe0e1b077eb18d11e0f6f27976cc","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183044,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[19,21],[2,21]],[[2,21],[19,21]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 19/21], [1, 19/21]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 19/21], [1, 19/21]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0045","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1645a9ee58f59f7bb31ecec3d4e718ceb868d09bf03477260b6348db80bd231a","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183045,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[10,11],[1,11]],[[1,11],[10,11]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 10/11], [1, 10/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 10/11], [1, 10/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0046","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cdae41a35ad661875da9a4cc9d8b6a40b02d7ea93f133c4ffd2c4f261d80978","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183046,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[21,23],[2,23]],[[2,23],[21,23]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 21/23], [1, 21/23]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 21/23], [1, 21/23]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0047","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea04566feef635d2389254e7d7ea271035d820638901f3f47dbf4bf475265321","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183047,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,12],[1,12]],[[1,12],[11,12]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 11/12], [1, 11/12]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 11/12], [1, 11/12]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0048","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87e3a47094f0ad5f89291e0bdfd3df88c8403e029ba7598e7b6ee0a24d0cc58d","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183048,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[23,25],[2,25]],[[2,25],[23,25]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 23/25], [1, 23/25]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 23/25], [1, 23/25]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0049","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684bfdaadb62684d693e9b133bcb837bbb39f361667d1c12d066c959625edecd","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183049,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[12,13],[1,13]],[[1,13],[12,13]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 12/13], [1, 12/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 12/13], [1, 12/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0050","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d78fe9a0ab3d5742b4fcef5d78e70412c00f375e18a5992962ea1a9e837e23b","family_id":"qf-mod49-ci-pointwise-coverage","seed":202611183050,"semantic_template":"ci-pointwise-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[25,27],[2,27]],[[2,27],[25,27]]],samples:[\"a\",\"b\"],theta:[0,1]}"},"fields":{"title":"Coverage Is a Repeated-Sampling Probability at Each Fixed Theta","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).","answer":{"kind":"expression","value":"([[0, 25/27], [1, 25/27]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)","explanation":"For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 25/27], [1, 25/27]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)."},"skills":["Coverage Is a Repeated-Sampling Probability at Each Fixed Theta"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0051","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4fb8e13fbf38f772d99df723f75cf8d75c3a8df14f1f34292a9c04ef7f58d45","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183051,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,3],[2,3]],[[2,3],[1,3]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 1/3], [1, 1/3]], 1/3, 1/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 1/3], [1, 1/3]], 1/3, 1/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0052","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360feff1cb76c4f92c2d272bbbe517c3d837ea6bfdb8a18156e566152897b993","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183052,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[1,2],[1,2]],[[1,2],[1,2]]],samples:[\"a\",\"b\"],target:[1,2],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 1/2], [1, 1/2]], 1/2, 1/2, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/2, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0053","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0559d8b9d81f7106b3a1c303955a7f8318ba2ded333e54dce5619dfecb9aaad0","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183053,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,5],[2,5]],[[2,5],[3,5]]],samples:[\"a\",\"b\"],target:[3,5],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 3/5], [1, 3/5]], 3/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 3/5], [1, 3/5]], 3/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0054","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d1a1d19dff4e9fd1aa08b684baf02844aac9ded709a8e8ccdb442d3bc09fb54","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183054,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[2,3],[1,3]],[[1,3],[2,3]]],samples:[\"a\",\"b\"],target:[2,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 2/3], [1, 2/3]], 2/3, 2/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 2/3], [1, 2/3]], 2/3, 2/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0055","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f241c39d800a24bd1ccbe83af74b654683fdf9727eea082c944579cd5c1938b","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183055,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,7],[2,7]],[[2,7],[5,7]]],samples:[\"a\",\"b\"],target:[5,7],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 5/7], [1, 5/7]], 5/7, 5/7, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 5/7], [1, 5/7]], 5/7, 5/7, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0056","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd7f6f2db77530ef27a34a2bacc6064dd5ec586412ba6a19d4d6d6553c51d74","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183056,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,4],[1,4]],[[1,4],[3,4]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 3/4], [1, 3/4]], 3/4, 1/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 3/4], [1, 3/4]], 3/4, 1/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0057","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f933291f573af7153a89e63826bc41b6f5674733ac307a03920c2610e728781c","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183057,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,9],[2,9]],[[2,9],[7,9]]],samples:[\"a\",\"b\"],target:[1,2],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 7/9], [1, 7/9]], 7/9, 1/2, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 7/9], [1, 7/9]], 7/9, 1/2, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0058","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eff175ce5f4fe7f54b0cbfa5269a6d609ee9fd600fc8e008e23dec86f4314b9b","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183058,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[4,5],[1,5]],[[1,5],[4,5]]],samples:[\"a\",\"b\"],target:[3,5],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 4/5], [1, 4/5]], 4/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 4/5], [1, 4/5]], 4/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0059","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ddfa04b43010c120c433e61433e1e4f1dfd5d46cdb2a9f9fb446a3ff9048f4c","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183059,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,11],[2,11]],[[2,11],[9,11]]],samples:[\"a\",\"b\"],target:[2,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 9/11], [1, 9/11]], 9/11, 2/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 9/11], [1, 9/11]], 9/11, 2/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0060","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f614bf44014c8a601dcea82ec6b3d782218c3d7878d41286c8c0446cd9b71aff","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183060,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[5,6],[1,6]],[[1,6],[5,6]]],samples:[\"a\",\"b\"],target:[5,7],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 5/6], [1, 5/6]], 5/6, 5/7, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 5/6], [1, 5/6]], 5/6, 5/7, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0061","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2946ff97740203b8710ef3dc35b4cf04817245be3b6f36d7aa10c6aedb1b08a4","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183061,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,13],[2,13]],[[2,13],[11,13]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 11/13], [1, 11/13]], 11/13, 1/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 11/13], [1, 11/13]], 11/13, 1/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0062","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1a32b3e03ed1b810b89b02516b856102a75b2d39ca8c2fe1b32db967aecffa2","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183062,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],target:[1,2],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 6/7], [1, 6/7]], 6/7, 1/2, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 6/7], [1, 6/7]], 6/7, 1/2, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0063","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8147593ff2c313c2b80f84e46eaba36ec0ebedbf25213cdbaf7abc58adff4f4","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183063,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[13,15],[2,15]],[[2,15],[13,15]]],samples:[\"a\",\"b\"],target:[3,5],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 13/15], [1, 13/15]], 13/15, 3/5, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 13/15], [1, 13/15]], 13/15, 3/5, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0064","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee9108644b266fc6c3a330c4d954a08b46c74e3ebbc8820526ae5062cc997de","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183064,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],target:[2,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 7/8], [1, 7/8]], 7/8, 2/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 7/8], [1, 7/8]], 7/8, 2/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0065","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1dfbaf4b01336a2c36f6b12cc3dce23f3ba0539f1ceafb688233bc96e950976","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183065,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[15,17],[2,17]],[[2,17],[15,17]]],samples:[\"a\",\"b\"],target:[5,7],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 15/17], [1, 15/17]], 15/17, 5/7, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 15/17], [1, 15/17]], 15/17, 5/7, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0066","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e0b9170cce7c80a2a87e9cebf242d2edce01aa4cf6d37f6b7a0ab71e5a1ccb9","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183066,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[8,9],[1,9]],[[1,9],[8,9]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 8/9], [1, 8/9]], 8/9, 1/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 8/9], [1, 8/9]], 8/9, 1/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0067","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b46c6123284ae03e756fe3c939d060b5cf9db350d582aaed1b729a779e1dfbcb","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183067,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[17,19],[2,19]],[[2,19],[17,19]]],samples:[\"a\",\"b\"],target:[1,2],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 17/19], [1, 17/19]], 17/19, 1/2, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 17/19], [1, 17/19]], 17/19, 1/2, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0068","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3224d9284e7ac6e1a272c6a2c447062422e2ffeb8e52ed971925d7955ad6e295","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183068,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[9,10],[1,10]],[[1,10],[9,10]]],samples:[\"a\",\"b\"],target:[3,5],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 9/10], [1, 9/10]], 9/10, 3/5, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 9/10], [1, 9/10]], 9/10, 3/5, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0069","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f854a57918364caebe7712b844526430ab3a50efd4fa0e06b1f07e1ae1493fd","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183069,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[19,21],[2,21]],[[2,21],[19,21]]],samples:[\"a\",\"b\"],target:[2,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 19/21], [1, 19/21]], 19/21, 2/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 19/21], [1, 19/21]], 19/21, 2/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0070","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c6fc5cbd5083b44d9bb379f3f1f4667a5b6f1229d1f374aea0d349609081af2","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183070,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[10,11],[1,11]],[[1,11],[10,11]]],samples:[\"a\",\"b\"],target:[5,7],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 10/11], [1, 10/11]], 10/11, 5/7, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 10/11], [1, 10/11]], 10/11, 5/7, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0071","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bcfb4e93b68d1a0289308751096454553fa9d9f1f03a968d4c7ccbd48077238","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183071,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[21,23],[2,23]],[[2,23],[21,23]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 21/23], [1, 21/23]], 21/23, 1/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 21/23], [1, 21/23]], 21/23, 1/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0072","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3cf4981ee52bb36aa3f11fc41aa3bac520a2bea2883f6846b14c531fa7e9e57","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183072,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,12],[1,12]],[[1,12],[11,12]]],samples:[\"a\",\"b\"],target:[1,2],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 11/12], [1, 11/12]], 11/12, 1/2, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 11/12], [1, 11/12]], 11/12, 1/2, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0073","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"442da4e493e1510824ebdc01436a43d7171e6bf781497a55bcf5df60ffa3a43d","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183073,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[23,25],[2,25]],[[2,25],[23,25]]],samples:[\"a\",\"b\"],target:[3,5],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 23/25], [1, 23/25]], 23/25, 3/5, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 23/25], [1, 23/25]], 23/25, 3/5, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0074","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6ef34408bb572574f849737e1cfb70e5b7fe0e01e1a0c60a635d78f7d9b15ea","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183074,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[12,13],[1,13]],[[1,13],[12,13]]],samples:[\"a\",\"b\"],target:[2,3],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 12/13], [1, 12/13]], 12/13, 2/3, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 12/13], [1, 12/13]], 12/13, 2/3, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0075","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"331e20dad2f7e7e672ddc2298df36b392872da7e67f74738ec1d6d80ea472543","family_id":"qf-mod49-ci-worst-case-coverage","seed":202611183075,"semantic_template":"ci-worst-case-coverage|{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[25,27],[2,27]],[[2,27],[25,27]]],samples:[\"a\",\"b\"],target:[5,7],theta:[0,1]}"},"fields":{"title":"A Uniform Coverage Target Is Checked against the Minimum, Not an Average","stem":"Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).","answer":{"kind":"expression","value":"([[0, 25/27], [1, 25/27]], 25/27, 5/7, Yes, worst_case_is_minimum_over_finite_theta)","explanation":"Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 25/27], [1, 25/27]], 25/27, 5/7, Yes, worst_case_is_minimum_over_finite_theta)."},"skills":["A Uniform Coverage Target Is Checked against the Minimum, Not an Average"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0076","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8030a6d82b13dc8f6688b5443af094f4772eee00aa9ca94be1fef8aafb3fabb0","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183076,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[1,3],[2,3]],[[2,3],[1,3]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[1,3],[2,3]],[[2,3],[1,3]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0077","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc2355a87fc7c6d01ac16bf2b090e793adc1d520c3bc49a7b83569825590a4eb","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183077,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[1,2],[1,2]],[[1,2],[1,2]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[2,4],[2,4]],[[2,4],[2,4]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/2, 1/2], [[0, 1/2], [1, 1/2]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/2, 1/2], [[0, 1/2], [1, 1/2]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0078","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd440245066f2ad6ff681010c14d7f68e7b5d5b4d4b9cfb3a50a6b6c6b73984f","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183078,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[3,5],[2,5]],[[2,5],[3,5]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[3,5],[2,5]],[[2,5],[3,5]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 3/5], [1, 3/5]], [[0, 3/5, 1], [1, 2/5, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 3/5], [1, 3/5]], [[0, 3/5, 1], [1, 2/5, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0079","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"140ddd1c8a9d12eb7bcc364c0d8fe589d142828c93ec1dac5d977d75d19a3c0a","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183079,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[4,1],[3,1]],[[4,1],[4,1]]],probability_rows:[[[2,3],[1,3]],[[1,3],[2,3]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[4,6],[2,6]],[[2,6],[4,6]]],\"intervals\":[[4,3],[4,4]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0080","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b15e56c2aab4ab733cd2c1c3f08035023fdb5e09e7581dd0ad0a14ee2e56b63","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183080,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[5,7],[2,7]],[[2,7],[5,7]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[5,7],[2,7]],[[2,7],[5,7]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0081","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab936f1e64d2e0ec97d7ab40fb82230a78135599399901a60793147c5a26264d","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183081,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[3,4],[1,4]],[[1,4],[3,4]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[6,8],[2,8]],[[2,8],[6,8]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/4, 3/4], [[0, 3/4], [1, 3/4]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/4, 3/4], [[0, 3/4], [1, 3/4]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0082","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6581d5b7b9cc5f3a8640c451184f5a9f044843bb7dfef9f95a6199ba0d2b0fee","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183082,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[7,9],[2,9]],[[2,9],[7,9]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[7,9],[2,9]],[[2,9],[7,9]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 7/9], [1, 7/9]], [[0, 7/9, 1], [1, 2/9, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 7/9], [1, 7/9]], [[0, 7/9, 1], [1, 2/9, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0083","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f296e706a4abd09d02660e1c81a3032a6a525e2ccfedd37868c5265cc95e4881","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183083,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[8,1],[7,1]],[[8,1],[8,1]]],probability_rows:[[[4,5],[1,5]],[[1,5],[4,5]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[8,10],[2,10]],[[2,10],[8,10]]],\"intervals\":[[8,7],[8,8]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0084","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f8e3eeb5e1a8dbb14596e6ff151afc0a079864d3965ddf2d33f242282be1bbd","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183084,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[9,11],[2,11]],[[2,11],[9,11]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[9,11],[2,11]],[[2,11],[9,11]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0085","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f73f54580f7a0f1235c02bcbb23f755d4e1bfd4434fb7516ea0ddf84006e5bfc","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183085,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[5,6],[1,6]],[[1,6],[5,6]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[10,12],[2,12]],[[2,12],[10,12]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/6, 5/6], [[0, 5/6], [1, 5/6]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/6, 5/6], [[0, 5/6], [1, 5/6]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0086","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f28e4288743b6c08cd433722594e4635fba341489722634698f1d98fc67c452","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183086,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[11,13],[2,13]],[[2,13],[11,13]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[11,13],[2,13]],[[2,13],[11,13]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 11/13], [1, 11/13]], [[0, 11/13, 1], [1, 2/13, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 11/13], [1, 11/13]], [[0, 11/13, 1], [1, 2/13, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0087","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"312836704f673a5f25f5d4ea208fd676c1794fb0c14262eabfb68bd293807f4a","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183087,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[12,1],[11,1]],[[12,1],[12,1]]],probability_rows:[[[6,7],[1,7]],[[1,7],[6,7]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[12,14],[2,14]],[[2,14],[12,14]]],\"intervals\":[[12,11],[12,12]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0088","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d860df4bd53125511f7571891ddd23a2035aab245dd0ebf41ad84c34bb27a0e9","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183088,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[13,15],[2,15]],[[2,15],[13,15]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[13,15],[2,15]],[[2,15],[13,15]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0089","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428b75f30b5692c9f8547db5dcaa8edefd878e22f231b19490941b6c17b9aed6","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183089,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[7,8],[1,8]],[[1,8],[7,8]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[14,16],[2,16]],[[2,16],[14,16]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/8, 7/8], [[0, 7/8], [1, 7/8]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/8, 7/8], [[0, 7/8], [1, 7/8]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0090","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e35481accc592c84cd5eddd7a239fa4af1f63726f717b4c1aba7fa9b30272cb","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183090,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[15,17],[2,17]],[[2,17],[15,17]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[15,17],[2,17]],[[2,17],[15,17]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 15/17], [1, 15/17]], [[0, 15/17, 1], [1, 2/17, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 15/17], [1, 15/17]], [[0, 15/17, 1], [1, 2/17, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0091","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aee5342fc9e3a137dba66e2ce5461d0741bbafd7a72a56ce2f163bd8097e344f","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183091,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[16,1],[15,1]],[[16,1],[16,1]]],probability_rows:[[[8,9],[1,9]],[[1,9],[8,9]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[16,18],[2,18]],[[2,18],[16,18]]],\"intervals\":[[16,15],[16,16]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0092","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff653132002787564b32e7ca9e049587d9d39737d4c7886ec65aca23af871174","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183092,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[17,19],[2,19]],[[2,19],[17,19]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[17,19],[2,19]],[[2,19],[17,19]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0093","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9849156ac6e2bc70ee871ab11a8a26c029ffdf30d2755ab571c919ff6313eb2b","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183093,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[9,10],[1,10]],[[1,10],[9,10]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[18,20],[2,20]],[[2,20],[18,20]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/10, 9/10], [[0, 9/10], [1, 9/10]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/10, 9/10], [[0, 9/10], [1, 9/10]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0094","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd364f274cdf8b93c0d240e7c7214dbbc1805a8fc083c39107646adb084616db","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183094,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[19,21],[2,21]],[[2,21],[19,21]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[19,21],[2,21]],[[2,21],[19,21]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 19/21], [1, 19/21]], [[0, 19/21, 1], [1, 2/21, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 19/21], [1, 19/21]], [[0, 19/21, 1], [1, 2/21, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0095","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ceb5961b6328e603497f96d94ead0b51971462eac875c4e876eeff9a6ade93","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183095,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[20,1],[19,1]],[[20,1],[20,1]]],probability_rows:[[[10,11],[1,11]],[[1,11],[10,11]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[20,22],[2,22]],[[2,22],[20,22]]],\"intervals\":[[20,19],[20,20]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0096","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f720731d092f5ef59fd435105cebff505a6465ebd2604a9c0e5bca1ea297cab","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183096,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[21,23],[2,23]],[[2,23],[21,23]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[21,23],[2,23]],[[2,23],[21,23]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0097","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eda73db6e99f322488f729f977d2aa1615a41ecd4a4fc9e7cfaf989e5f05b768","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183097,"semantic_template":"ci-interpretation-boundary|{mode:\"posterior_confusion\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],observed:\"b\",prior:[[1,2],[1,2]],probability_rows:[[[11,12],[1,12]],[[1,12],[11,12]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=posterior_confusion and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[22,24],[2,24]],[[2,24],[22,24]]],\"intervals\":[[0,0],[1,1]],\"prior\":[[2,4],[2,4]],\"observed\":\"b\"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).","answer":{"kind":"expression","value":"(posterior_confusion, b, [1/12, 11/12], [[0, 11/12], [1, 11/12]], No, posterior_given_observed_data_is_not_frequentist_coverage)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/12, 11/12], [[0, 11/12], [1, 11/12]], No, posterior_given_observed_data_is_not_frequentist_coverage)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0098","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c70befd1bc8b2de28b16fc495fd81e7fce7cf3783378e49b6c6442fa71a9e5db","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183098,"semantic_template":"ci-interpretation-boundary|{mode:\"ignored_selection\",model:{intervals:[[[0,1],[0,1]],[[1,1],[1,1]]],probability_rows:[[[23,25],[2,25]],[[2,25],[23,25]]],samples:[\"a\",\"b\"],selected_samples:[\"a\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=ignored_selection and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[23,25],[2,25]],[[2,25],[23,25]]],\"intervals\":[[0,0],[1,1]],\"selected_samples\":[\"a\"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).","answer":{"kind":"expression","value":"(ignored_selection, [a], [[0, 23/25], [1, 23/25]], [[0, 23/25, 1], [1, 2/25, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 23/25], [1, 23/25]], [[0, 23/25, 1], [1, 2/25, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0099","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00d9997f51be8a5bbb377ebd3f7b44a646f6e0521e96088278ce6d90039d3a4f","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183099,"semantic_template":"ci-interpretation-boundary|{mode:\"invalid_procedure\",model:{intervals:[[[24,1],[23,1]],[[24,1],[24,1]]],probability_rows:[[[12,13],[1,13]],[[1,13],[12,13]]],samples:[\"a\",\"b\"],theta:[0,1]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=invalid_procedure and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[24,26],[2,26]],[[2,26],[24,26]]],\"intervals\":[[24,23],[24,24]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).","answer":{"kind":"expression","value":"(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0100","batch_id":"batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f21d476771595f981eb11382b17bb56ab9fade06ff75b4ff545fcb021bf89105","family_id":"qf-mod49-ci-interpretation-boundary","seed":202611183100,"semantic_template":"ci-interpretation-boundary|{mode:\"average_only\",model:{intervals:[[[0,1],[0,1]],[[0,1],[0,1]]],probability_rows:[[[25,27],[2,27]],[[2,27],[25,27]]],samples:[\"a\",\"b\"],target:[1,3],theta:[0,1],theta_weights:[[1,3],[2,3]]}}"},"fields":{"title":"Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities","stem":"Use mode=average_only and complete model={\"theta_values\":[0,1],\"samples\":[\"a\",\"b\"],\"probability_rows\":[[[25,27],[2,27]],[[2,27],[25,27]]],\"intervals\":[[0,0],[0,0]],\"theta_weights\":[[1,3],[2,3]],\"target\":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).","answer":{"kind":"expression","value":"(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)","explanation":"average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)."},"skills":["Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities"],"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."}}