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AP Statistics 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions- Exam Style Questions - MCQs - New Syllabus

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Question 

You may notice that “plain math” questions tend to be easier than word problems. Consider these two questions:

I.    What is 3.5% of 2,000?

II.  A nurse needs to add some salt to a bag of water in the amount of 3.5% of the weight of the bag. If the bag weighs 2,000 grams, how much salt must the nurse add?

In a simple random sample of 90 U.S. adults, 72 answered Question I correctly. In another simple random sample of 180 U.S. adults, 108 answered Question II correctly. A 90% confidence interval for \( p_I – p_{II} \) is \( (0.108,\;0.292) \). Do these data provide convincing statistical evidence that the proportion of all U.S. adults who would get the simpler question correct is higher than the proportion of U.S. Adults who would get the word problem correct?

(A) No, because the confidence interval is above 0.05.
(B) No, because the confidence interval is not above 0.50.
(C) Yes, because 0.20, the difference between the sample proportions, is in the confidence interval.
(D) Yes, because the confidence interval is entirely above 0.
(E) Yes, because 0.292 is greater than 0.108.

▶️ Answer/Explanation

The confidence interval for the difference in population proportions is:
\( (0.108,\;0.292) \)

Because the entire interval is positive and does not include \(0\), all plausible values for
\( p_I – p_{II} \)
are greater than zero.

This provides convincing evidence that the population proportion who would answer the simpler question correctly is higher than the population proportion who would answer the word problem correctly.

When a confidence interval for a difference excludes zero, it indicates a statistically significant difference between the two population proportions at the corresponding confidence level.

✅ Answer: (D)

Question 

Independent random samples of 100 luxury cars and 250 non-luxury cars in a certain city are examined to see if they have bumper stickers. Of the 250 non-luxury cars, 125 have bumper stickers and of the 100 luxury cars, 30 have bumper stickers. Which of the following is a 90 percent confidence interval for the difference in the proportion of non-luxury cars with bumper stickers and the proportion of luxury cars with bumper stickers from the population of cars represented by these samples?

(A) \( (0.5-0.3)\pm1.645\sqrt{\frac{(0.5)(0.5)}{250}+\frac{(0.3)(0.7)}{100}} \)
(B) \( (0.5-0.3)\pm1.96\sqrt{\frac{(0.5)(0.5)}{250}+\frac{(0.3)(0.7)}{100}} \)
(C) \( (0.5-0.3)\pm1.645\sqrt{\left(\frac{155}{350}\right)\left(\frac{195}{350}\right)\left(\frac{1}{250}+\frac{1}{100}\right)} \)
(D) \( (0.5-0.3)\pm1.96\sqrt{\left(\frac{155}{350}\right)\left(\frac{195}{350}\right)\left(\frac{1}{250}+\frac{1}{100}\right)} \)
(E) \( (0.5-0.3)\pm1.645\sqrt{(0.4)(0.6)\left(\frac{1}{250}+\frac{1}{100}\right)} \)

▶️ Answer/Explanation

First calculate the sample proportions:
\( \hat{p}_1=\frac{125}{250}=0.5 \)
\( \hat{p}_2=\frac{30}{100}=0.3 \)
For a 90% confidence interval for the difference of two population proportions, the formula is:
\( (\hat{p}_1-\hat{p}_2)\pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+ \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \)
Since this is a 90% confidence interval:
\( z^*=1.645 \)
Substituting the values gives exactly choice (A). Notice that confidence intervals use the individual sample proportions in the standard error rather than a pooled proportion.

✅ Answer: (A)

Question 

A polling organization asks a random sample of 1,000 registered voters which of two candidates they plan to vote for in an upcoming election. Candidate A is preferred by 400 respondents, Candidate B is preferred by 500 respondents, and 100 respondents are undecided. George uses a large sample confidence interval for two proportions to estimate the difference in the population proportions favoring the two candidates. This procedure is not appropriate because

(A) the two sample proportions were not computed from independent samples
(B) the sample size was too small
(C) the third category, undecided, makes the procedure invalid
(D) the sample proportions are different; therefore the variances are probably different as well
(E) George should have taken the difference \( \frac{500-400}{1000} \) and then used a large sample confidence interval for a single proportion instead

▶️ Answer/Explanation

A confidence interval for the difference between two population proportions requires two independent samples or two independently selected groups.

In this study, the same sample of 1,000 voters was used to classify respondents into the categories Candidate A, Candidate B, or Undecided. Because the proportions for Candidates A and B come from the same sample, the observations are not independent.

If one respondent is counted as favoring Candidate A, that respondent cannot simultaneously be counted as favoring Candidate B. This dependence violates the assumptions required for a two-sample confidence interval for proportions.

The issue is not the sample size or the presence of an undecided category. The primary problem is the lack of independence between the two estimated proportions.

✅ Answer: (A)

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