MATH1127: Introduction to Statistics (60022) Lesson 8.2 A Single Population Mean using the Student t-Distribution
A researcher must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:
\begin{tabular}{|r|}
\hline 25.9 \\
\hline 33.5 \\
\hline 38.1 \\
\hline 31.2 \\
\hline 44.9 \\
\hline 41.1 \\
\hline 20.5 \\
\hline 44.6 \\
\hline 4.6 \\
\hline 57.1 \\
\hline
\end{tabular}
Assuming the population of temperatures are normally distributed, find the $99 \%$ confidence interval. Round the answers to two decimal places. Enter your answer as an interval of the form (LB,UP).
The researcher is $99 \%$ confident that the population average temperature is within the temperature interval

Step 1 ：Given the sample temperatures, we need to estimate the mean temperature with a 99% confidence interval. The temperatures are as follows: 25.9, 33.5, 38.1, 31.2, 44.9, 41.1, 20.5, 44.6, 4.6, 57.1.

Step 2 ：First, we calculate the sample mean (\(\bar{x}\)) and the sample standard deviation (s). The sample mean is the sum of all the sample temperatures divided by the number of samples. The sample standard deviation is the square root of the variance, which is the average of the squared differences from the mean.

Step 3 ：The sample mean (\(\bar{x}\)) is 34.15 and the sample standard deviation (s) is 14.73.

Step 4 ：Next, we need to find the t-score for the 99% confidence level. The t-score is a value that allows us to calculate the confidence interval. It depends on the desired level of confidence and the sample size. For a 99% confidence level and a sample size of 10, the t-score is 3.25.

Step 5 ：Finally, we can calculate the confidence interval using the formula \(\bar{x} \pm t \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, t is the t-score, s is the sample standard deviation, and n is the sample size.

Step 6 ：The lower bound of the confidence interval is 19.01 and the upper bound is 49.29.

Step 7 ：\(\boxed{\text{Final Answer: The 99% confidence interval for the population average temperature is (19.01, 49.29).}}\)