Part 4 of 4 points Points: 0 of 1 Save You are given the sample mean and the population standard deviation. Use this information to construct the $90 \%$ and $95 \%$ confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 72 dates, the mean record high daily temperature in a certain city has a mean of $82.22^{\circ} \mathrm{F}$. Assume the population standard deviation is $14.91^{\circ} \mathrm{F}$. (Round to two decimal places as needed.) Which interval is wider? Choose the correct answer below. The $95 \%$ confidence interval The $90 \%$ confidence interval Interpret the results. A. You can be $90 \%$ confident that the population mean record high temperature is between the bounds of the $90 \%$ confidence interval, and $95 \%$ confident for the $95 \%$ interval. B. You can be certain that the population mean record high temperature is either between the lower bounds of the $90 \%$ and $95 \%$ confidence intervals or the upper bounds of the $90 \%$ and $95 \%$ confidence intervals. C. You can be certain that the mean record high temperature was within the $90 \%$ confidence interval for approximately 65 of the 72 days, and was within the $95 \%$ confidence interval for approximately 68 of the 72 days. D. You can be $90 \%$ confident that the population mean record high temperature is outside the bounds of the $90 \%$ confidence interval, and $95 \%$ confident for the $95 \%$ interval.
Solution
Step 1 ：Given the sample mean \(\bar{x} = 82.22\), the population standard deviation \(\sigma = 14.91\), and the sample size \(n = 72\).
Step 2 ：The formula for a confidence interval is \(\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\), where \(Z\) is the Z-score, which depends on the confidence level.
Step 3 ：For a 90% confidence level, the Z-score is approximately 1.645. So, the 90% confidence interval is \(82.22 \pm 1.645 \frac{14.91}{\sqrt{72}}\), which is approximately \((79.33, 85.11)\).
Step 4 ：For a 95% confidence level, the Z-score is approximately 1.96. So, the 95% confidence interval is \(82.22 \pm 1.96 \frac{14.91}{\sqrt{72}}\), which is approximately \((78.78, 85.66)\).
Step 5 ：The width of a confidence interval is simply the upper bound minus the lower bound. The width of the 90% confidence interval is approximately \(85.11 - 79.33 = 5.78\), and the width of the 95% confidence interval is approximately \(85.66 - 78.78 = 6.89\).
Step 6 ：\(\boxed{\text{The 95% confidence interval is wider.}}\)
Step 7 ：The correct interpretation of the results is: You can be 90% confident that the population mean record high temperature is between the bounds of the 90% confidence interval, and 95% confident for the 95% interval. This is because the confidence level represents the probability that the population mean lies within the confidence interval.
