(CO 4) In a sample of 15 stuffed animals, you find that they weigh an average of 8.56 ounces with a standard deviation of 0.06 ounces. Find the $92 \%$ confidence interval. $(8.527,8.593)$ $(8.526,8.594)$ $(8.531,8.589)$ $(8.52 \times, 8.599)$
Solution
Step 1 ：Given values are: sample mean \(x_{bar} = 8.56\), standard deviation \(s = 0.06\), sample size \(n = 15\), and Z-score for 92% confidence level \(Z = 1.75\).
Step 2 ：Calculate the margin of error using the formula: \(Z \times \frac{s}{\sqrt{n}}\).
Step 3 ：Substitute the given values into the formula to get the margin of error: \(1.75 \times \frac{0.06}{\sqrt{15}} = 0.027110883423451915\).
Step 4 ：Calculate the confidence interval using the formula: \(x_{bar} \pm\) margin of error.
Step 5 ：Substitute the values into the formula to get the confidence interval: \(8.56 - 0.027110883423451915 = 8.532889116576548\) and \(8.56 + 0.027110883423451915 = 8.587110883423453\).
Step 6 ：The 92% confidence interval is \((8.532889116576548, 8.587110883423453)\), which can be approximated to \((8.533, 8.587)\).
Step 7 ：Comparing the calculated confidence interval with the given options, it is closest to the option \((8.531,8.589)\).
Step 8 ：Final Answer: The 92% confidence interval is \(\boxed{(8.531,8.589)}\).
