Constructing Confidence Intervals for p1-p2 You can construct ...

Question

Constructing Confidence Intervals for p1-p2 You can construct a confidence interval for the difference between two population proportions p1-p2 by using the inequality below.

$({\hat{p}}_{1} - {\hat{p}}_{2}) - z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}} < p_{1} - p_{2} < ({\hat{p}}_{1} - {\hat{p}}_{2}) + z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}}$

In Exercises 23–26, construct the indicated confidence interval for p1-p2. Assume the samples are random and independent.

Students Planning to Study Visual and Performing Arts In a survey of 10,000 students taking the SAT, 7% were planning to study visual and performing arts in college. In another survey of 8000 students taken 10 years before, 8% were planning to study visual and performing arts in college. Construct a 95% confidence interval for p1-p2, where p1 is the proportion from the recent survey and p2 is the proportion from the survey taken 10 years ago. (Adapted from College Board)

Accepted Answer

All right, hello, everyone. So this question says, a poll of 7500 high school seniors found that 12% plan to attend a community college. Five years earlier, a poll of 6200, excuse me, 6200 seniors showed a 10% plan to attend a community college. Construct a 95% confidence interval for P1 subtracted by P2, where P1 is the recent proportion and P2 is the earlier proportion. Here we have 4 different answer choices labeled A through D. All right, so first, let's identify what we know. First, we know that Phat one, that's the sample proportion for the Recent group is 0.12. Whereas N1, the sample size is 7500. Now for the earlier group, he had 2. Is equal to. 0.10 And N2 is equal to 6200. So now we find the difference in the sample proportions. So Phat1 subtracted by PAT 2. Equals 0.12 subtracted by 0.10, which gives you 0.02. Now, what you want to do is find the standard error for the difference. So standard error is equal to The square root of And all of what I'm about to write is under the square root to so clear. So it is He had one multiplied by one subtracted by P had 1. Divided by N1. Added to He had 2 multiplied by 1 subtracted by Pha 2. And divided by n 2. So now plugging in the information that you have, the standard error is equal to 0.12. Multiplied by 0.88. Divided by 7500. And added to 0.10 multiplied by 0.90. And divided by 6200. And this equals approximately 0.00535. Now, for the 95% confidence interval, what you want to find is the critical value. So Z for the critical value is equal to 1.96. Which leads you now to the margin of error. Now, the margin of error is equal to the critical Z value, that's 1.96, multiplied by your standard error. So 1.96 multiplied. By 0.00535 Is approximately equal to 0.01049. So now you're left with your confidence interval. Now the confidence interval. Is going to be equal to. Your difference in sample proportions, that's 0.02, added to and subtracted by your margin of error. So 0.01049. So ultimately After rounding to four decimal places, the lower bound of your confidence interval is 0.0095, whereas the upper bound is 0.0305. And there you have it. Ultimately, the correct answer is going to be option C in the multiple choice. And so with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Confidence Interval

Proportion and Sample Size

Z-score and Standard Error

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