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AP Statistics 7.8 Setting Up a Test for the Difference of Two Population Means - MCQs - Exam Style Questions

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Question

To obtain certification for a certain occupation, candidates take a proficiency exam. The exam consists of two sections, and neither section should be more difficult than the other. To investigate whether one section of the exam was more difficult than the other, a random sample of 50 candidates was selected. The candidates took the exam and their scores on each section were recorded. The table shows the summary statistics.
 Mean Percent CorrectStandard Deviation Percent Correct
First section7510
Second section655
Difference108
Which of the following is the test statistic for the appropriate test to determine if there is a significant mean difference between the percent correct on the two sections (first minus second) for all candidates similar to those in the investigation?
(A) $t=\frac{75-65}{\frac{8}{\sqrt{50}}}$ 
(B) $t=\frac{75-65}{\sqrt{\frac{10^{2}}{50}+\frac{5^{2}}{50}}}$
(C) $\chi^{2}=\frac{(75-70)^{2}}{70}+\frac{(65-70)^{2}}{70}$ 
(D) $\chi^{2}=\frac{(75-70)^{2}}{75}+\frac{(65-70)^{2}}{65}$ 
(E) $z=\frac{0.75-0.65}{\sqrt{0.7(1-0.7)(\frac{1}{50}+\frac{1}{50})}}$ 
▶️ Answer/Explanation
Detailed solution

1. Identify the Test Type:
Since the same 50 candidates took both sections of the exam, the data are paired. The appropriate test is a **paired t-test** for the mean difference. 

2. State the Formula:
The test statistic for a paired t-test is:
$t = \frac{\bar{x}_{diff}}{s_{diff} / \sqrt{n}}$
where $\bar{x}_{diff}$ is the mean of the differences, $s_{diff}$ is the standard deviation of the differences, and $n$ is the sample size.

3. Substitute Values from the Table:
– $\bar{x}_{diff} = 10$ 
– $s_{diff} = 8$
– $n = 50$

The mean difference, 10, is also the difference between the mean scores ($75 – 65$).  So, the numerator can be written as $75-65$.

4. Construct the Test Statistic:
$t = \frac{75-65}{\frac{8}{\sqrt{50}}}$

This matches option (A).
✅ Answer: (A)

Question

A two-sided t-test for a population mean is conducted of the null hypothesis \(H_{0}: \mu=100\). If a 90 percent t-interval constructed from the same sample data contains the value of 100, which of the following can be concluded about the test at a significance level of \(\alpha=0.10\)?
(A) The p-value is less than 0.10, and \(H_{0}\) should be rejected.
(B) The p-value is less than 0.10, and \(H_{0}\) should not be rejected.
(C) The p-value is greater than 0.10, and \(H_{0}\) should be rejected.
(D) The p-value is greater than 0.10, and \(H_{0}\) should not be rejected.
(E) There is not enough information given to make a conclusion about the p-value and \(H_{0}\).
▶️ Answer/Explanation
Detailed solution

A \(90\%\) confidence interval corresponds to a two-sided hypothesis test with a significance level of \(\alpha = 1 – 0.90 = 0.10\).

The rule for this duality is: if the hypothesized value (\(\mu=100\)) is contained within the confidence interval, then we fail to reject the null hypothesis. Failing to reject \(H_0\) means the p-value is greater than the significance level \(\alpha\).

Therefore, the p-value is greater than \(0.10\), and \(H_0\) should not be rejected.
✅ Answer: (D)

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