A researcher must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:
\begin{tabular}{|r|}
\hline 68.5 \\
\hline 67.6 \\
\hline 68.5 \\
\hline 84.3 \\
\hline 81.8 \\
\hline 86.5 \\
\hline 64.7 \\
\hline 61 \\
\hline 80.3 \\

Assuming the population of temperatures are normally distributed, find the 80%confidence interval. Round the answers to two decimal places. Enter your answer as an interval of the form (LB,UP).
The researcher is 80%confident that the population average temperature is within the temperature interval

Step 1 ：Given a set of sample temperatures in degrees Fahrenheit: 68.5, 67.6, 68.5, 84.3, 81.8, 86.5, 64.7, 61, 80.3.

Step 2 ：We are asked to find the 80% confidence interval for the mean temperature, assuming the population of temperatures are normally distributed.

Step 3 ：To find the 80% confidence interval for the mean temperature, we first calculate the sample mean and the standard error of the mean.

Step 4 ：The sample mean is calculated as the sum of all sample temperatures divided by the number of samples. In this case, the sample mean is approximately 73.69.

Step 5 ：The standard error of the mean is the standard deviation of the sample divided by the square root of the sample size. In this case, the standard error of the mean is approximately 3.16.

Step 6 ：The 80% confidence interval is then the sample mean plus and minus the standard error of the mean times the z-score for an 80% confidence interval. The z-score for an 80% confidence interval is approximately 1.28.

Step 7 ：Using these values, we can calculate the 80% confidence interval for the mean temperature as (69.64, 77.74).

Step 8 ：Final Answer: The researcher is 80% confident that the population average temperature is within the temperature interval \(\boxed{(69.64, 77.74)}\).