Out of 168 randomly selected adults in the United States who were surveyed, 80 exercise on a regular basis. Construct a $95 \%$ confidence interval for the proportion of all adults in the United States who exercise on a regular basis. Round to three decimal places.
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G 2023 Hawkes Learning
Answer
Calculating the above expression, we find that the 95% confidence interval for the proportion of all adults in the United States who exercise on a regular basis is \(\boxed{[0.401, 0.552]}\).
Step 1 ：Given that the sample size (n) is 168 and the number of successes (people who exercise regularly) is 80, we first calculate the sample proportion (p_hat) which is the number of successes divided by the sample size. This gives us \(p_{hat} = \frac{80}{168} = 0.476\).
Step 2 ：We then use the Z-score corresponding to the desired level of confidence, which for a 95% confidence interval is approximately 1.96.
Step 3 ：Next, we calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{p_{hat}(1-p_{hat})}{n}} = \sqrt{\frac{0.476(1-0.476)}{168}} = 0.039\).
Step 4 ：Finally, we substitute p_hat, Z, and SE into the formula for the confidence interval, which gives us \(CI = p_{hat} \pm Z*SE = 0.476 \pm 1.96*0.039\).
Step 5 ：Calculating the above expression, we find that the 95% confidence interval for the proportion of all adults in the United States who exercise on a regular basis is \(\boxed{[0.401, 0.552]}\).
