The high temperatures (in degrees Fahrenheit) of a random sample of 6 small towns are: \[ \begin{array}{|c|} \hline 96.6 \\ \hline 96.7 \\ \hline 98.7 \\ \hline 98.6 \\ \hline 99.4 \\ \hline 98.8 \\ \hline \end{array} \] Assume high temperatures are normally distributed. Based on this data, find the $80 \%$ confidence interval of the mean high temperature of towns. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place). \[ 80 \% \text { C.I. }= \] Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places. Question Help: $\square$ Video 1 Video 2 D post to forum

Step 1 ：Given the high temperatures of a random sample of 6 small towns are: 96.6, 96.7, 98.7, 98.6, 99.4, 98.8.

Step 2 ：We assume high temperatures are normally distributed.

Step 3 ：We need to find the 80% confidence interval of the mean high temperature of towns.

Step 4 ：First, we calculate the mean of the sample data. The mean is given by \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\), where \(x_i\) are the data points and \(n\) is the number of data points. For our data, the mean is approximately 98.13.

Step 5 ：Next, we calculate the standard deviation of the sample data. The standard deviation is given by \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\), where \(x_i\) are the data points, \(\bar{x}\) is the mean, and \(n\) is the number of data points. For our data, the standard deviation is approximately 1.18.

Step 6 ：We use the formula for the confidence interval of the mean, which is given by \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level (for 80%, \(z\) is approximately 1.282), \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 7 ：Substituting the values into the formula, we get the lower and upper bounds of the confidence interval as approximately 97.51 and 98.75 respectively.

Step 8 ：\(\boxed{\text{Final Answer: The 80% confidence interval for the mean high temperature of towns is approximately }(97.51, 98.75)}\)