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AP Statistics 7.5 Carrying Out a Test for a Population Mean - MCQs - Exam Style Questions

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Question

According to a report, the mean number of rolls of toilet paper used each year by residents of a certain city is 100. To investigate this claim, researchers surveyed a random sample of residents from the city and recorded the number of rolls of toilet paper they used each year. They used the data to test the hypotheses \(H_{0}:\mu=100\) versus \(H_{a}:\mu\neq 100\), where \(\mu\) is the mean number of rolls of toilet paper used each year by all residents of the city.
The conditions for inference were met and the p-value of the test is 0.058. At the significance level of \(\alpha=0.05\), which of the following statements is true?
(A) There is convincing statistical evidence that the mean number of rolls of toilet paper used each year by all residents of the city is less than 100.
(B) There is convincing statistical evidence that the mean number of rolls of toilet paper used each year by all residents of the city is different from 100.
(C) The researchers should reject that the mean number of rolls of toilet paper used each year by all residents of the city is 100.
(D) There is not convincing statistical evidence that the mean number of rolls of toilet paper used each year by all residents of the city is different from 100.
(E) The researchers failed to obtain any evidence that the mean number of rolls of toilet paper used each year in the city is different from 100.
▶️ Answer/Explanation
Detailed solution
Compare the p-value to \(\alpha\): \(0.058 > 0.05\).
Because \(p > \alpha\), we fail to reject \(H_0\). The data do not provide convincing evidence that \(\mu\) differs from 100.
(E) overstates the conclusion (“no evidence”); we only say “not convincing evidence.”
✅ Answer: (D)

Question

A state study on labor reported that one-third of full-time teachers in the state also worked part time at another job. For those teachers, the average number of hours worked per week at the part-time job was \(13\). After an increase in state teacher salaries, a random sample of \(400\) teachers who worked part time at another job was selected. The average number of hours worked per week at the part-time job for the teachers in the sample was \(12.5\) with standard deviation \(6.5\) hours. Is there convincing statistical evidence, at the level of \(\alpha=0.05\), that the average number of hours worked per week at part-time jobs decreased after the salary increase?
(A) No. The \(p\)-value of the appropriate test is greater than \(0.05\).
(B) No. The \(p\)-value of the appropriate test is less than \(0.05\).
(C) Yes. The \(p\)-value of the appropriate test is greater than \(0.05\).
(D) Yes. The \(p\)-value of the appropriate test is less than \(0.05\).
(E) Not enough information is given to determine whether there is convincing statistical evidence.
▶️ Answer/Explanation
Detailed solution

Hypotheses (one-sample \(t\) for a mean): \(H_0:\mu=13\) vs. \(H_a:\mu<13\).
Test statistic: \[ t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}} =\frac{12.5-13}{6.5/\sqrt{400}} =\frac{-0.5}{6.5/20} \approx \frac{-0.5}{0.325}\approx -1.54. \] With \(df=n-1=399\), the one-sided \(p\)-value is about \(0.062\), which is \(>0.05\).
Therefore we fail to reject \(H_0\); there is not convincing evidence that the mean weekly hours decreased after the salary increase.
✅ Answer: (A)

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