AP Statistics 6.8 Confidence Intervals for the Difference of Two Proportions- FRQs - Exam Style Questions
▶️ Answer/Explanation
(a)
State: We want to construct and interpret a \(95\%\) confidence interval for \(p\), the true proportion of all adults in the United States who would have chosen the economy statement.
Plan: The appropriate procedure is a one-sample z-interval for a population proportion. The problem states that the conditions for inference have been met , so we do not need to check them.
Do: The sample proportion is \(\hat{p} = 0.37\).The sample size is \(n = 1,048\) For a \(95\%\) confidence level, the critical value is \(z^* = 1.96\).
The confidence interval is calculated using the formula: [\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] \[ 0.37 \pm 1.96 \sqrt{\frac{(0.37)(1-0.37)}{1,048}} \] \[ 0.37 \pm 1.96 \sqrt{\frac{(0.37)(0.63)}{1,048}} \] \[ 0.37 \pm 1.96 \sqrt{0.000222} \] \[ 0.37 \pm 1.96 (0.0149) \] \[ 0.37 \pm 0.0292 \] The interval is \((0.3408, 0.3992)\).
Conclude: We are \(95\) percent confident that the interval from \(0.34\) to \(0.40\) captures the true proportion of all adults in the United States who would have chosen the economy statement.
(b)
This condition, often called the Large Counts condition (\(n\hat{p} \ge 10\) and \(n(1-\hat{p}) \ge 10\)), is necessary to ensure that the sampling distribution of the sample proportion, \(\hat{p}\), is approximately normal. The formula used to calculate the z-confidence interval relies on the normal approximation to the binomial distribution. This approximation is reasonably accurate only when the expected number of successes and failures are both sufficiently large (at least \(10\)) .
(c)
No, the two-sample z-interval for a difference between proportions is not an appropriate procedure.
Justification: This procedure requires the two proportions to come from two independent random samples. In this study, there was only one random sample of \(1,048\) adults. The proportions of adults choosing the environment statement (\(58\%\)) and the economy statement (\(37\%\)) were calculated from the responses of the same group of people. Therefore, the two proportions are not independent; they are dependent because they come from the same sample, and the choice of one statement affects whether the other could be chosen.