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AP Statistics 7.4 Setting Up a Test for a Population Mean - MCQs - Exam Style Questions

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Question

A researcher at a company with a large number of employees wanted to investigate whether a training course would improve employees’ understanding of the company policies. The researcher selected a random sample of 9 employees from the company and gave each selected employee an exam prior to the training to identify their understanding of the company policies. Each selected employee was given a score on the exam (from 0—no understanding to 100—excellent understanding). After the employees completed the training course, they were given the same exam. The results are summarized in the table.
Results of the Exams
 Sample SizeMeanStandard Deviation
Score before training course932.217.6
Score after training course962.114.6
Difference (after minus before) of score929.915.7
Assume the conditions for inference have been met. Which of the following is closest to the value of the test statistic of the appropriate test for investigating whether there is a mean difference in exam score (after minus before) for all employees of the company?
(A) \(t=1.90\)
(B) \(t=3.92\)
(C) \(t=4.95\)
(D) \(t=5.71\)
(E) \(t=12.76\)
▶️ Answer/Explanation
Detailed solution
This is a matched-pairs (one-sample \(t\) on the differences) test.
Test statistic: \(t=\dfrac{\bar d-0}{s_d/\sqrt{n}}\).
Here \(\bar d=29.9\), \(s_d=15.7\), \(n=9\).
Compute: \(t=\dfrac{29.9}{15.7/\sqrt{9}}=\dfrac{29.9}{15.7/3}=\dfrac{29.9}{5.233\ldots}\approx 5.71\).
✅ Answer: (D)

Question

Two high schools have a similar number of students and parking lots of similar size. The safety officers at both schools want to investigate whether there is an average difference in the number of cars parked per day in the student parking lots for the school year. A random sample of \(15\) school days will be selected. For each selected day, the number of cars parked in the student parking lots will be counted at both schools and the difference will be recorded. Assuming all conditions for inference are met, which of the following is the appropriate test for the investigation?
(A) A two-sample \(z\)-test for a difference between proportions
(B) A two-sample \(t\)-test for a difference between means
(C) A matched-pairs \(t\)-test for a mean difference
(D) A chi-square test of homogeneity
(E) A chi-square test of independence
▶️ Answer/Explanation
Detailed solution

For each sampled day, counts from the two schools are taken on the same day, so observations are naturally paired by day. We compute the daily differences (School A − School B), yielding \(n=15\) paired differences and test whether the population mean difference is \(0\). The correct procedure is a matched-pairs \(t\)-test on the mean of the differences.
Two-sample tests (A,B) assume independent samples, which is not the design here. Chi-square tests (D,E) are for categorical data, not quantitative counts with pairing.
✅ Answer: (C)

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