Stealing Automobiles A random sample of 32 months has a mean of 58 automobiles stolen per month. The population standard deviation was 3 automobiles obtained from a long-term study. Find the $92 \%$ confidence interval for the mean. Round intermediate calculations to two decimal places and your final answers to the nearest whole number.
$< \mu< $

Step 1 ：Given in the problem, we have: \(\bar{x} = 58\) (mean number of automobiles stolen per month), \(\sigma = 3\) (population standard deviation), and \(n = 32\) (sample size).

Step 2 ：We want to find the $92\%$ confidence interval, so \(\alpha = 1 - 0.92 = 0.08\). Therefore, \(Z_{\alpha/2} = Z_{0.04}\).

Step 3 ：Looking up \(Z_{0.04}\) in the Z-table, we find that \(Z_{0.04} \approx 1.75\).

Step 4 ：Substituting these values into the formula, we get: \(58 \pm 1.75 * \frac{3}{\sqrt{32}}\).

Step 5 ：Calculating the expression inside the parentheses: \(\frac{3}{\sqrt{32}} \approx 0.53\) (rounded to two decimal places).

Step 6 ：So, the confidence interval is: \(58 \pm 1.75 * 0.53\).

Step 7 ：Calculating the interval: \(58 - 1.75 * 0.53 \approx 57\) (rounded to the nearest whole number) and \(58 + 1.75 * 0.53 \approx 59\) (rounded to the nearest whole number).

Step 8 ：So, the $92\%$ confidence interval for the mean number of automobiles stolen per month is from $57$ to $59$. Therefore, the final answer is \(\boxed{[57, 59]}\).