{"id":"cb_question_bank:4a8e6334-f2f2-47e0-932b-501a5e2ae178","source_id":"cb_question_bank","source_item_id":"4a8e6334-f2f2-47e0-932b-501a5e2ae178","external_id":"a4fc06ae-2b85-4830-a27c-e248eebde1f9","ibn":null,"question_id":"f8a322d9","program":"SAT","module":"math","domain_code":"Q","domain":"Problem-Solving and Data Analysis","skill_code":"Q.C.","skill":"One-variable data: Distributions and measures of center and spread","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Data Set A: The horizontal axis is labeled Integer. It ranges from 10 to 60 and is divided into 5 equal intervals. The vertical axis is labeled Frequency. It ranges from 0 to 12 in increments of 1, with values marked every 2 grid lines. The histogram has a skewed right shape. The histogram has 4 bins. The Frequency data for the 4 bins are as follows: 20 to 30: 3 30 to 40: 4 40 to 50: 7 50 to 60: 9 Data Set B: The horizontal axis is labeled Integer. It ranges from 10 to 60 and is divided into 5 equal intervals. The vertical axis is labeled Frequency. It ranges from 0 to 12 in increments of 1, with values marked every 2 grid lines. The histogram has a skewed right shape. The histogram has 4 bins. The Frequency data for the 4 bins are as follows: 10 to 20: 3 20 to 30: 4 30 to 40: 7 40 to 50: 9 Two data sets of 23 integers each are summarized in the histograms shown. For each of the histograms, the first interval represents the frequency of integers greater than or equal to 10 , but less than 20 . The second interval represents the frequency of integers greater than or equal to 20 , but less than 30 , and so on. What is the smallest possible difference between the mean of data set A and the mean of data set B?","stimulus":"","stimulus_html":null,"stem_html":"

\n

Two data sets of 23\n integers each are summarized in the histograms shown. For each of the histograms, the first interval represents the frequency of integers greater than or equal to 10\n, but less than 20\n. The second interval represents the frequency of integers greater than or equal to 20\n, but less than 30\n, and so on. What is the smallest possible difference between the mean of data set A and the mean of data set B?

","rationale_html":"

Choice B is correct. The histograms shown have the same shape, but data set A contains values between 20\n and 60\n and data set B contains values between 10\n and 50\n. Thus, the mean of data set A is greater than the mean of data set B. Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is the difference between the smallest possible mean of data set A and the greatest possible mean of data set B. In data set A, since there are 3\n integers in the interval greater than or equal to 20\n but less than 30\n, 4\n integers greater than or equal to 30\n but less than 40\n, 7\n integers greater than or equal to 40\n but less than 50\n, and 9\n integers greater than or equal to 50\n but less than 60\n, the smallest possible mean for data set A is 3·20+4·30+7·40+9·5023. In data set B, since there are 3\n integers greater than or equal to 10\n but less than 20\n, 4\n integers greater than or equal to 20\n but less than 30\n, 7\n integers greater than or equal to 30\n but less than 40\n, and 9\n integers greater than or equal to 40\n but less than 50\n, the largest possible mean for data set B is 3·19+4·29+7·39+9·4923. Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is 3·20+4·30+7·40+9·5023-3·19+4·29+7·39+9·4923, which is equivalent to 3·20-3·19+4·30-4·29+7·40-7·39+9·50-9·4923. This expression can be rewritten as 320-19+430-29+740-39+950-4923, or 2323, which is equal to 1\n. Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is 1\n.

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Choice A is incorrect. This is the smallest possible difference between the ranges, not the means, of the data sets.

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Choice C is incorrect. This is the difference between the greatest possible mean, not the smallest possible mean, of data set A and the greatest possible mean of data set B.

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Choice D is incorrect. This is the smallest possible difference between the sum of the values in data set A and the sum of the values in data set B, not the smallest possible difference between the means.

","correct_answer":["B"],"has_media":true,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.833000+00:00","source_updated_at":"2023-08-02T20:25:59.833000+00:00","options":[{"label":"A","content_html":"

0\n

","ord":0},{"label":"B","content_html":"

1\n

","ord":1},{"label":"C","content_html":"

10\n

","ord":2},{"label":"D","content_html":"

23\n

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}