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AP Statistics 4.9 Setting Up a Test for the Difference Between Two Population Mean - Exam Style Questions - MCQs - New Syllabus

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Question 

A survey was conducted with a random sample of 250 U.S. adults who had school-age children and 300 U.S. adults who did not have school-age children. Each adult reported their current salary. The average salary for the 250 adults who had school-age children was $\$54,420$ and the average salary for the 300 adults who did not have school-age children was $\$63,050$.

Research Question: Is the population mean salary different for people who have school-age children versus those not having school-age children?

What is the correct alternative hypothesis statement?

(A) \(\mu_1 \ne \mu_2\) where 1 = people who have school-age children and 2 = people who do not have school-age children
(B) \(\mu_1 = \mu_2\) where 1 = people who have school-age children and 2 = people who do not have school-age children
(C) \(\mu_1 < \mu_2\) where 1 = people who have school-age children and 2 = people who do not have school-age children
(D) \(p_1 \ne p_2\) where 1 = people who have school-age children and 2 = people who do not have school-age children
(E) \(p_1 < p_2\) where 1 = people who have school-age children and 2 = people who do not have school-age children

▶️ Answer/Explanation

The research question asks whether the population mean salary is different between two groups. Because the parameter of interest is a mean, the hypotheses must use \(\mu\), not \(p\).

The phrase “is different” indicates a two-sided test, so the alternative hypothesis should allow for either group to have the larger mean.

Therefore, the correct alternative hypothesis is:

\(H_A:\mu_1 \ne \mu_2\)

where group 1 consists of adults with school-age children and group 2 consists of adults without school-age children.

✅ Answer: (A)

Question

Stefan conducted a study to compare reading comprehension scores for children who read a story at \(9\text{ a.m.}\) and children who read the same story at \(3\text{ p.m.}\). A total of \(100\) children volunteered to participate. Fifty children were randomly assigned to the \(9\text{ a.m.}\) group, and the remaining \(50\) children were randomly assigned to the \(3\text{ p.m.}\) group.
Table 1: Summary Statistics of Reading Scores
 \(n\)MeanStandard Deviation
\(9\text{ a.m.}\)\(50\)\(15.2\)\(4.12\)
\(3\text{ p.m.}\)\(50\)\(17.9\)\(4.43\)
Which of the following best explains why a two-sample \(t\)-test for the difference in two population means is more appropriate than a paired \(t\)-test?
(A) The two groups are independent because different children were assigned to the \(9\text{ a.m.}\) and \(3\text{ p.m.}\) groups.
(B) The two groups are paired because all children answered the same \(25\) questions.
(C) The two groups are paired because both groups had the same sample size, \(n=50\).
(D) The two groups are independent because the two sample means are different.
▶️ Answer/Explanation

A paired \(t\)-test is used when each observation in one group has a natural matching observation in the other group, such as the same child tested twice or matched pairs of children.
In this study, different children were assigned to the \(9\text{ a.m.}\) and \(3\text{ p.m.}\) groups. A child in the \(9\text{ a.m.}\) group is not naturally paired with a particular child in the \(3\text{ p.m.}\) group.
Therefore, the two samples are independent, so a two-sample \(t\)-test for the difference in two population means is appropriate.

✅ Answer: (A)

Question 

A study compared the language skills and mental development of two groups of 24-month-old children. One group consisted of children identified as talkative, and the other group consisted of children identified as quiet. The scores for the two groups on a test that measured language skills are shown in the table below.

Assuming that it is reasonable to regard the groups as simple random samples and that the other conditions for inference are met, what statistical test should be used to determine if there is a significant difference in the average test score of talkative and quiet children at the age of 24 months?
(A) A chi-square goodness-of-fit test
(B) A chi-square test of independence
(C) A matched-pairs \(t\)-test for means
(D) A two-sample \(t\)-test for means
(E) A linear regression \(t\)-test
▶️ Answer/Explanation
Detailed solution

1. Identify Data Type:
We have quantitative data (test scores) from two independent groups.

2. Appropriate Test:
To compare means from two independent groups, use two-sample t-test.

3. Why Other Tests are Incorrect:
(A) & (B) are for categorical data
(C) is for paired/dependent data
(E) is for relationship between two quantitative variables

✅ Answer: (D)

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