To test the effect of a physical fitness course on one's physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. Ten individuals are randomly selected to participate in the course. The results are displayed in the following table. Using this data, find the $95 \%$ confidence interval for the true difference in the number of sit-ups each person can do before and after the course. Assume that the numbers of sit-ups are normally distributed for the population both before and after completing the course.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline Sit-ups before & 42 & 42 & 23 & 32 & 30 & 42 & 25 & 47 & 35 & 38 \\
\hline Sit-ups after & 57 & 48 & 29 & 41 & 36 & 57 & 40 & 51 & 41 & 40 \\
\hline
\end{tabular}
Step 2 of 4: Calculate the sample standard deviation of the paired differences. Round your answer to six decimal places.
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Final Answer: The sample standard deviation of the paired differences is \(\boxed{4.880801}\).
Step 1 ：The problem asks for the sample standard deviation of the paired differences. This means we need to calculate the difference between the number of sit-ups before and after the course for each individual, and then calculate the standard deviation of these differences.
Step 2 ：First, we list the number of sit-ups each individual could do before the course: \(42, 42, 23, 32, 30, 42, 25, 47, 35, 38\).
Step 3 ：Next, we list the number of sit-ups each individual could do after the course: \(57, 48, 29, 41, 36, 57, 40, 51, 41, 40\).
Step 4 ：We then calculate the difference between the number of sit-ups before and after the course for each individual, resulting in the following list: \(15, 6, 6, 9, 6, 15, 15, 4, 6, 2\).
Step 5 ：Finally, we calculate the sample standard deviation of these differences, which is approximately \(4.880801\).
Step 6 ：Final Answer: The sample standard deviation of the paired differences is \(\boxed{4.880801}\).
