In a random sample of 15 -year-olds, the mean height was 65 inches. The sampling distribution was normal with a standard error of 0.50 inches. A. [ 3 points] Construct a \( 90 \% \) confidence interval to estimate the population mean. Round your answer to 3 decimal places. Use the general form of a confidence interval: sample statistic \( \pm z^{*}(S E) \) Show your work.

**Step 1: Identify the given information** - Sample mean: \( \bar{x} = 65 \) inches - Standard error: \( SE = 0.50 \) inches - Confidence level: \( 90\% \) **Step 2: Find the critical value \( z^{*} \) for a \( 90\% \) confidence level** For a \( 90\% \) confidence interval, the tail probability is \[ \alpha = 1 - 0.90 = 0.10 \] Since the confidence interval is symmetric, each tail has a probability of \[ \frac{\alpha}{2} = 0.05. \] The critical value \( z^{*} \) corresponds to the \( 95^{th} \) percentile of the standard normal distribution, which is \[ z^{*} \approx 1.645. \] **Step 3: Compute the margin of error** The margin of error is calculated as: \[ ME = z^{*} \times SE = 1.645 \times 0.50 = 0.8225. \] **Step 4: Construct the confidence interval** The general form of the confidence interval is: \[ \bar{x} \pm z^{*}(SE) \] Substitute the known values: \[ 65 \pm 0.8225 \] Thus, the lower bound is: \[ 65 - 0.8225 = 64.1775, \] and the upper bound is: \[ 65 + 0.8225 = 65.8225. \] **Step 5: Round the answer to 3 decimal places** Lower bound: \[ 64.1775 \approx 64.178 \] Upper bound: \[ 65.8225 \approx 65.823. \] The \( 90\% \) confidence interval to estimate the population mean is: \[ [64.178, \ 65.823]. \]