Consider the following random sample of diameter measurements (in inches) of 16 softballs.
\[
\begin{array}{l}
\text { 4.82, 4.7, 4.84, 4.75, 4.68, 4.87, 4.78, 4.9, 4.71, 4.83, 4.81, 4.86, 4.69, 4.72, 4.87, } 4.84 \\
\text { Send data to calculator Send data to Excel }
\end{array}
\]
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If we assume that the diameter measurements are normally distributed, find a $90 \%$ confidence interval for the mean diameter of a softball. Give the lower limit and upper limit of the $90 \%$ confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)
Lower limit: []
Upper limit: $\square$
S

Step 1 ：Given the diameter measurements of 16 softballs, we are asked to find a 90% confidence interval for the mean diameter assuming the measurements are normally distributed.

Step 2 ：First, we calculate the sample mean and the standard error of the mean. The standard error of the mean is the standard deviation divided by the square root of the sample size.

Step 3 ：The sample mean is calculated as \( \frac{1}{n} \sum_{i=1}^{n} x_i \), where \( x_i \) are the diameter measurements and \( n \) is the sample size. In this case, \( n = 16 \) and the diameter measurements are [4.82, 4.7, 4.84, 4.75, 4.68, 4.87, 4.78, 4.9, 4.71, 4.83, 4.81, 4.86, 4.69, 4.72, 4.87, 4.84]. The sample mean is calculated to be 4.791875.

Step 4 ：The standard deviation is calculated as \( \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \), where \( \bar{x} \) is the sample mean. The standard deviation is calculated to be 0.07341378163079015.

Step 5 ：The standard error of the mean is then calculated as \( \frac{\text{standard deviation}}{\sqrt{n}} \), which is 0.01835344540769754.

Step 6 ：The confidence interval is then the sample mean plus and minus the product of the standard error and the z-score corresponding to the desired level of confidence. The z-score for a 90% confidence interval is approximately 1.645.

Step 7 ：The lower limit of the confidence interval is calculated as \( \text{mean} - \text{z-score} \times \text{standard error} \), which is 4.761686268754093.

Step 8 ：The upper limit of the confidence interval is calculated as \( \text{mean} + \text{z-score} \times \text{standard error} \), which is 4.822063731245907.

Step 9 ：Rounding to two decimal places, the lower limit of the 90% confidence interval for the mean diameter of a softball is \(\boxed{4.76}\) inches and the upper limit is \(\boxed{4.82}\) inches.