A researcher must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures: \begin{tabular}{|r|} \hline 27.6 \\ \hline-14.3 \\ \hline 0.6 \\ \hline 43.9 \\ \hline 2.4 \\ \hline 27.7 \\ \hline 18.1 \\ \hline-4.1 \\ \hline 27.1 \\ \hline 4.5 \\ \hline-6 \\ \hline 32.9 \\ \hline \end{tabular} Assuming the population of temperatures are normally distributed, find the $98 \%$ confidence interval. Round the answers to two decimal places. Enter your answer as an interval of the form (LB,UP). The researcher is $98 \%$ confident that the population average temperature is within the temperature interval

Step 1 ：Given a set of sample temperatures, we are asked to find the 98% confidence interval for the mean temperature. The temperatures are: 27.6, -14.3, 0.6, 43.9, 2.4, 27.7, 18.1, -4.1, 27.1, 4.5, -6, 32.9.

Step 2 ：The sample size (n) is 12, as there are 12 temperatures.

Step 3 ：The sample mean (\(\bar{x}\)) is calculated by adding all the sample temperatures and dividing by the sample size. The sample mean is approximately 13.37 degrees Fahrenheit.

Step 4 ：The sample standard deviation (s) is calculated using the formula for standard deviation. The sample standard deviation is approximately 18.44.

Step 5 ：For a 98% confidence interval, the Z-score (Z) is 2.33.

Step 6 ：We can now calculate the confidence interval using the formula \(\bar{x} \pm Z \frac{s}{\sqrt{n}}\).

Step 7 ：The lower bound of the confidence interval is calculated as \(\bar{x} - Z \frac{s}{\sqrt{n}}\), which is approximately 0.98 degrees Fahrenheit.

Step 8 ：The upper bound of the confidence interval is calculated as \(\bar{x} + Z \frac{s}{\sqrt{n}}\), which is approximately 25.75 degrees Fahrenheit.

Step 9 ：Final Answer: The researcher is 98% confident that the population average temperature is within the temperature interval \(\boxed{(0.98, 25.75)}\) degrees Fahrenheit.