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.

Sit-ups before	52	22	35	31	53	35	47	29	22	44
Sit-ups after	57	33	42	46	56	46	53	46	38	49

Step 2 of 4 :  Calculate the sample standard deviation of the paired differences. Round your answer to six decimal places.

To find the sample standard deviation of the paired differences between sit-ups before and after the course, follow these steps:

1. **Calculate the differences** for each individual:
   \[
   \text{Difference} = \text{Sit-ups After} - \text{Sit-ups Before}
   \]

   Using the provided dataTo find the sample standard deviation of the paired differences between sit-ups before and after the course, follow these steps:

1. **Calculate the differences** for each individual:
   \[
   \text{Difference} = \text{Sit-ups After} - \text{Sit-ups Before}
   \]

   Using the provided 
   - Differences:
     - \(57 - 52 = 5\)
     - \(33 - 22 = 11\)
     - \(42 - 35 = 7\)
     - \(46 - 31 = 15\)
     - \(56 - 53 = 3\)
     - \(46 - 35 = 11\)
     - \(53 - 47 = 6\)
     - \(46 - 29 = 17\)
     - \(38 - 22 = 16\)
     - \(49 - 44 = 5\)

   Thus, the differences are:
   \[
   5, 11, 7, 15, 3, 11, 6, 17, 16, 5
   \]

2. **Calculate the mean of the differences** (\(\bar{d}\)):
   \[
   \bar{d} = \frac{5 + 11 + 7 + 15 + 3 + 11 + 6 + 17 + 16 + 5}{10} = \frac{  5 + 11 + 7 + 15 + 3 + 11 + 6 + 17 + 16 + 5}{10} = \frac{  11 + 11 + 7 + 15 + 3 + 11 + 6 + 17 + 16 + 5}{10} = \frac{  5 + 11 + 11 + 15 + 3 + 11 + 6 + 17 + 16 + 5}{10} = \frac{  100}{10} = 10
   \]

3. **Calculate the squared differences from the mean**:
   \[
   (d_i - \bar{d})^2
   \]
   - For \(5: (5 - 10)^2 = 25\)
   - For \(11: (11 - 10)^2 = 1\)
   - For \(7: (7 - 10)^2 = 9\)
   - For \(15: (15 - 10)^2 = 25\)
   - For \(3: (3 - 10)^2 = 49\)
   - For \(11: (11 - 10)^2 = 1\)
   - For \(6: (6 - 10)^2 = 16\)
   - For \(17: (17 - 10)^2 = 49\)
   - For \(16: (16 - 10)^2 = 36\)
   - For \(5: (5 - 10)^2 = 25\)

4. **Sum the squared differences**:
   \[
   25 + 1 + 9 + 25 + 49 + 1 + 16 + 49 + 36 + 25 = 236
   \]

5. **Calculate the sample variance**:
   \[
   s^2 = \frac{\text{Sum of squared differences}}{n - 1} = \frac{236}{10 - 1} = \frac{236}{9} \approx 26.222222
   \]

6. **Calculate the sample standard deviation** (\(s\)):
   \[
   s = \sqrt{26.222222} \approx 5.121
   \]

Rounding to six decimal places, the sample standard deviation of the paired differences is:
\[
s \approx 5.121000
\]

### Summary:
The sample standard deviation of the paired differences is approximately **5.121000**.

Would you like more details or have any questions about this calculation? Here are some follow-up questions:
1. How would you calculate the mean of the paired differences?
2. What is the next step to find the 95% confidence interval?
3. Can you explain the significance of the paired t-test?
4. How would the confidence interval change if the sample size increased?
5. What assumptions are necessary for this analysis? 

**Tip:** Always ensure your data meets the assumptions of normality when performing parametric tests like the paired t-test.

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. 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. Calculate the sample standard deviation of the paired differences.

Explore the step-by-step process of calculating the sample standard deviation of paired differences in sit-ups before and after a fitness course. This problem involves analyzing data from ten individuals, with results indicating significant differences in performance. Ideal for high school statistics students.

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