You measure 38 randomly selected textbooks' weights, and find they have a mean weight of 68 ounces. Assume the population standard deviation is 8.7 ounces. Based on this, construct a $99 \%$ confidence interval for the true population mean textbook weight.

Give your answers to 2 decimal places.
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Step 1 ：We are given that the sample mean (x_bar) of the weights of 38 randomly selected textbooks is 68 ounces, and the population standard deviation (sigma) is 8.7 ounces. We are asked to construct a 99% confidence interval for the true population mean textbook weight.

Step 2 ：The first step is to calculate the z-score for the given confidence level. The z-score is a measure of how many standard deviations an element is from the mean. In this case, we use the formula for the z-score of a normal distribution, which is \(z = \Phi^{-1}\left(\frac{1 + \text{confidence level}}{2}\right)\), where \(\Phi^{-1}\) is the inverse of the cumulative distribution function of the standard normal distribution. Substituting the given confidence level of 0.99 into the formula, we get \(z = \Phi^{-1}\left(\frac{1 + 0.99}{2}\right) = 2.5758293035489004\).

Step 3 ：Next, we calculate the margin of error, which is the range in which we expect the true population mean to lie with a certain level of confidence. The formula for the margin of error is \(E = z \cdot \frac{\sigma}{\sqrt{n}}\), where \(z\) is the z-score, \(\sigma\) is the population standard deviation, and \(n\) is the sample size. Substituting the given values into the formula, we get \(E = 2.5758293035489004 \cdot \frac{8.7}{\sqrt{38}} = 3.635335804844095\).

Step 4 ：Finally, we calculate the confidence interval, which is the range in which we expect the true population mean to lie with a certain level of confidence. The formula for the confidence interval is \(\text{lower limit} = x_{\text{bar}} - E\) and \(\text{upper limit} = x_{\text{bar}} + E\), where \(x_{\text{bar}}\) is the sample mean and \(E\) is the margin of error. Substituting the given values into the formula, we get \(\text{lower limit} = 68 - 3.635335804844095 = 64.36466419515591\) and \(\text{upper limit} = 68 + 3.635335804844095 = 71.63533580484409\).

Step 5 ：Rounding to two decimal places, the 99% confidence interval for the true population mean textbook weight is \(\boxed{64.36 < \mu < 71.64}\) ounces.