AP Statistics 3.12 Setting Up a Test for the Difference Between Two Population Proportions- Exam Style Questions - MCQs - New Syllabus

Question

Employees were randomly sampled from two large corporations. Out of 68 randomly selected employees from Corporation A, 58 were happy with their job. Out of 74 randomly selected employees from Corporation B, 52 were happy with their job. A 95 percent confidence interval was computed as follows:

\(\left(\frac{58}{68}-\frac{52}{74}\right)\pm1.96\sqrt{\frac{\frac{58}{68}\left(1-\frac{58}{68}\right)}{68}+\frac{\frac{52}{74}\left(1-\frac{52}{74}\right)}{74}}=[0.016,\;0.284]\)

Which of the following is an appropriate interpretation of the 95 percent confidence interval?

(A) We are 95% confident the proportion of employees happy with their job at both corporations is between 0.016 and 0.284.
(B) We are 95% confident the sample proportion of employees happy with their job at both corporations is between 0.016 and 0.284.
(C) We are 95% confident the difference in population proportions of employees happy with their job at Corporation A and Corporation B is between 0.016 and 0.284
(D) We are 95% confident the difference in the number of employees happy with their job at Corporation A and Corporation B is between 0.016 and 0.284.
(E) We are 95% confident the proportion of 68 employees happy with their job at Corporation A minus the proportion of 74 employees happy with their job at Corporation B is between 0.016 and 0.284.

▶️ Answer/Explanation

The confidence interval was constructed using:

\(\hat{p}_A-\hat{p}_B\)

which estimates the difference between the population proportions of employees who are happy with their jobs at the two corporations.

A confidence interval should always be interpreted in terms of the population parameter being estimated, not the sample statistics used to compute it.

Therefore, the correct interpretation is that we are 95% confident the true difference in population proportions of employees happy with their jobs at Corporation A and Corporation B lies between 0.016 and 0.284.

Because the entire interval is positive, it suggests that the proportion of happy employees is higher at Corporation A than at Corporation B.

✅ Answer: (C)

Question

A large exercise center has several thousand members from age \(18\) to \(55\) years and several thousand members age \(56\) and older. The manager is investigating whether members’ opinions of taking online fitness classes differ by age.

The manager selected a random sample of \(170\) exercise center members ages \(18\) to \(55\) years and a second random sample of \(230\) exercise center members ages \(56\) years and older. Of those sampled, \(51\) of the \(170\) younger members and \(79\) of the \(230\) older members said they would be interested in taking online fitness classes.

Let \(p_{\text{younger}}\) represent the true proportion of all exercise center members ages \(18\) to \(55\) years who would be interested in taking online fitness classes. Let \(p_{\text{older}}\) represent the true proportion of all exercise center members ages \(56\) years and older who would be interested in taking online fitness classes.

Which of the following gives the correct hypotheses for testing whether the two age groups differ in their interest in taking online fitness classes?

(A) \(H_0:p_{\text{younger}}=p_{\text{older}}\), \(H_a:p_{\text{younger}}\neq p_{\text{older}}\)
(B) \(H_0:p_{\text{younger}}=p_{\text{older}}\), \(H_a:p_{\text{younger}}>p_{\text{older}}\)
(C) \(H_0:\hat{p}_{\text{younger}}=\hat{p}_{\text{older}}\), \(H_a:\hat{p}_{\text{younger}}\neq\hat{p}_{\text{older}}\)
(D) \(H_0:p_{\text{younger}}=0.30\), \(H_a:p_{\text{older}}=0.3435\)

▶️ Answer/Explanation

The manager is investigating whether the proportions differ by age, so the alternative hypothesis should be two-sided.
The hypotheses should be written in terms of the population proportions, not the sample proportions.
Therefore, the correct hypotheses are:
\(H_0:p_{\text{younger}}=p_{\text{older}}\)
\(H_a:p_{\text{younger}}\neq p_{\text{older}}\)

✅ Answer: (A)

Question

The Interstate Highway Act of 1957 designated a scheme for numbering of the Interstate highways in the United States. A recent survey was taken asking people in the United States if they could explain the difference between an odd-numbered highway and an even-numbered highway. In a randomly selected sample of people aged 40 and older, 203 out of 240 were able to explain the difference. In a randomly selected sample of people under the age of 40, 188 out of 240 were able to explain the difference. Based on these samples, is there a significant difference in the proportion of people in the United States aged 40 and older and people under age 40 who can explain the difference between odd-numbered routes and even-numbered routes in the US Interstate Highway numbering system?

Note: Odd-numbered routes generally run north to south and even-numbered routes generally run east to west.

(A) Since the test statistic is 1.8 which is greater than 1, we can conclude there is a statistical difference between the two age groups (40 and older versus under 40).
(B) Since 0.04 < 0.05, we can conclude there is a statistical difference between the two age groups (40 and older versus under 40).
(C) Since 0.06 > 0.05, we can conclude there is insufficient evidence of a statistical difference between the two age groups (40 and older versus under 40).
(D) Since 0.08 > 0.05, we can conclude there is a statistical difference between the two age groups (40 and older versus under 40).
(E) Since 0.08 > 0.05, we can conclude there is insufficient evidence of a statistical difference between the two age groups (40 and older versus under 40).

▶️ Answer/Explanation

First compute the sample proportions:

\(\hat{p}_1=\frac{203}{240}=0.8458\)
\(\hat{p}_2=\frac{188}{240}=0.7833\)

The difference in sample proportions is:

\(\hat{p}_1-\hat{p}_2=0.0625\)

A two-proportion z-test for these data gives a p-value of approximately 0.08. Since:

\(0.08 > 0.05\)

we fail to reject the null hypothesis. There is not enough evidence to conclude that the population proportions differ between the two age groups.
✅ Answer: (E)

AP Statistics 3.12 Setting Up a Test for the Difference Between Two Population Proportions- Exam Style Questions - MCQs - New Syllabus

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