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AP Statistics 3.13 Carrying Out a Test for the Difference Between Two Population Proportions- Exam Style Questions - MCQs - New Syllabus

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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 selected a random sample of \(170\) members ages \(18\) to \(55\) years and a second random sample of \(230\) members ages \(56\) years and older.
The manager found that \(51\) of the \(170\) sampled members ages \(18\) to \(55\) years and \(79\) of the \(230\) sampled members ages \(56\) years and older said they would be interested in taking online fitness classes.
For a two-sample \(z\)-test for the difference in population proportions, which of the following correctly gives the combined sample proportion \(\hat{p}_c\) used to check the large counts condition?
(A) \(\hat{p}_c=\dfrac{51}{170}=0.30\)
(B) \(\hat{p}_c=\dfrac{79}{230}\approx0.3435\)
(C) \(\hat{p}_c=\dfrac{51+79}{170+230}=0.325\)
(D) \(\hat{p}_c=\dfrac{0.30+0.3435}{2}\approx0.3218\)
▶️ Answer/Explanation

For a test of \(H_0:p_{\text{younger}}=p_{\text{older}}\), the large counts condition and standard error use the combined sample proportion.
The combined sample proportion is calculated by combining the successes and combining the total sample sizes:
\(\hat{p}_c=\dfrac{51+79}{170+230}\)
\(\hat{p}_c=\dfrac{130}{400}=0.325\)
This value is then used to calculate expected successes and failures for both samples.

✅ Answer: (C)

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)

Question 

Two types of medication were given to different patients to determine if there is a difference in the proportion of adults who showed a reduction in headache symptoms within 30 minutes of taking each medication. Eighteen out of the randomly sampled 300 adults given medication A did not see a reduction in their headache symptoms within 30 minutes. Twenty-four out of another random sample of 300 adults given medication B did not see a reduction in headache symptoms within 30 minutes of taking the medication.

Which of the following is the correct test statistic to test if there is statistical evidence of a difference in the effectiveness between Medication A and Medication B?

(A) \(\frac{0.94-0.92}{\sqrt{0.94(0.06)\left(\frac{2}{600}\right)}}\)

(B) \(\frac{0.94-0.92}{\sqrt{0.08(0.92)\left(\frac{1}{300}+\frac{1}{300}\right)}}\)

(C) \(\frac{0.08-0.06}{\sqrt{0.06(0.92)\left(\frac{2}{600}\right)}}\)

(D) \(\frac{0.94-0.92}{\sqrt{0.06(0.94)\left(\frac{1}{300}+\frac{1}{300}\right)}}\)

(E) \(\frac{0.94-0.92}{\sqrt{0.93(0.07)\left(\frac{1}{300}+\frac{1}{300}\right)}}\)

▶️ Answer/Explanation

For Medication A, the proportion showing improvement is:

\(\hat{p}_1=\frac{300-18}{300}=\frac{282}{300}=0.94\)

For Medication B, the proportion showing improvement is:

\(\hat{p}_2=\frac{300-24}{300}=\frac{276}{300}=0.92\)

For a two-proportion z-test, use the pooled proportion:

\(\hat{p}=\frac{282+276}{600}=\frac{558}{600}=0.93\)

\(\hat{q}=1-\hat{p}=0.07\)

The test statistic is:

\(z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}\hat{q}\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}\)

\(\frac{0.94-0.92}{\sqrt{0.93(0.07)\left(\frac{1}{300}+\frac{1}{300}\right)}}\)

This matches choice (E).
✅ Answer: (E)

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