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. If convenient, use technology to construct the confidence intervals.
A random sample of 45 home theater systems has a mean price of $\$ 139.00$. Assume the population standard deviation is $\$ 15.10$.
Construct a $90 \%$ confidence interval for the population mean.
The $90 \%$ confidence interval is
(Round to two decimal places as needed.)

Step 1 ：We are given the sample mean (\(\bar{x}\)) as $139.00, the population standard deviation (\(\sigma\)) as $15.10, and the sample size (\(n\)) as 45. We are asked to construct a $90\%$ confidence interval for the population mean.

Step 2 ：The formula for a confidence interval is \(\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\), where \(Z\) is the Z-score which corresponds to the desired confidence level.

Step 3 ：The Z-score for a $90\%$ confidence interval is approximately $1.645$. This value can be found in a standard Z-table or using a calculator.

Step 4 ：Substituting these values into the formula, we get \(139.00 \pm 1.645 \frac{15.10}{\sqrt{45}}\).

Step 5 ：Calculating the margin of error, we get approximately $3.70.

Step 6 ：Subtracting and adding this margin of error from the sample mean, we get the confidence interval as approximately \((135.30, 142.70)\).

Step 7 ：Final Answer: The $90\%$ confidence interval for the population mean is \(\boxed{(135.30, 142.70)}\). This means we are $90\%$ confident that the true population mean lies within this interval.