AP Statistics 3.10 Constructing a Confidence Interval for the Difference Between Two Population Proportions Questions

Practise choosing, checking, and calculating two-sample z-intervals for differences between population proportions, with group order kept explicit.

Exam points

choose a two-sample z-interval to estimate the contextual difference between two population proportions
reject a two-proportion interval when both response categories come from the same people and are dependent
construct p-hat1 minus p-hat2 plus or minus z-star times the unpooled standard error

Covered syllabus points

3.10.A—Identify an appropriate confidence interval procedure including the parameters for the difference between two…3.10.B—Justify the appropriateness of constructing a confidence interval for the difference between two population…3.10.C—Calculate an appropriate confidence interval for the difference between two population proportions 3.10.D—Calculate the standard error and margin of error for estimating the difference between two population…

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The campaign manager for a candidate for governor wants to estimate the difference between the percentage of voters who viewed an advertisement and favor the candidate and the percentage of voters who had not viewed the advertisement and favor the candidate. Two independent random samples of 500 voters were selected. One random sample consisted of voters who viewed the advertisement and the other random sample consisted of voters who had not viewed the advertisement. The table shows the frequencies of responses for those surveyed.

Which of the following is the most appropriate method for the campaign manager to use to estimate the difference between the percentages of voters who favor the candidate?

A one-sample z-interval for estimating a population proportion

A t-interval for estimating the slope of the least-squares regression line

A matched-pairs t-interval for estimating a mean difference

A two-sample z-interval for estimating a difference between sample proportions

A two-sample z-interval for estimating a difference between population proportions

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35. In a random sample of 1,500 college students, a pollster found 45\% prefer a female president and $42 \%$ prefer a male president. To calculate a 95 percent confidence interval for the difference in the proportion of college students who prefer a female president over a male president, he uses $(0.45-0.42) \pm 1.96 \sqrt{\frac{(0.45)(0.55)}{1,500}+\frac{(0.42)(0.58)}{1,500}}$ . Were conditions for inference met?

Yes, because there was a random sample, 1,500 is less than 10\% of all college students, and 1,500(0.45), 1,500(0.55), 1,500(0.42), and 1,500(0.58) are all $\geq 10$ .

No, because there was no random assignment between female and male college students.

No, because the independence assumption is violated.

No, because the random sample may not be truly representative of the population.

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Two brands of golf cart batteries, R and S, were tested to investigate whether there is a difference between the proportions of fully charged batteries that could power a golf cart for at least thirty miles. A random sample of 50 brand R batteries was selected and a random sample of 50 brand S batteries was selected. Batteries were randomly assigned to one of 100 different golf carts. Each cart was driven until the battery failed. Of the carts with brand R batteries, 40 traveled over thirty miles, and of the carts with brand S batteries, 35 traveled over thirty miles. Which of the following is a 95 percent confidence interval for the difference in the proportions of fully charged batteries that can power a golf cart for at least thirty miles for the two brands?

Statistics 2021Intl MCQ

$(0.8-0.7) \pm 1.645 \sqrt{\frac{(0.8)(0.2)}{50}+\frac{(0.7)(0.3)}{50}}$

$(0.8-0.7) \pm 1.645 \sqrt{(0.75)(0.25)\left(\frac{1}{50}+\frac{1}{50}\right)}$

$(0.8-0.7) \pm 1.645 \sqrt{\frac{(0.8)(0.2)}{50}-\frac{(0.7)(0.3)}{50}}$

$(0.8-0.7) \pm 1.960 \sqrt{\frac{(0.8)(0.2)}{50}+\frac{(0.7)(0.3)}{50}}$

$(0.8-0.7) \pm 1.960 \sqrt{(0.75)(0.25)\left(\frac{1}{50}+\frac{1}{50}\right)}$

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