The mean of the differences is 193 points, and the standard deviation of the differences is 62.73 points. The conditions for inference are met. What is the correct [tex]$98\%$[/tex] confidence interval for the mean difference (after - before) in score?

A. [tex]$193 \pm 2.764\left(\frac{62.73}{\sqrt{9}}\right)$[/tex]

B. [tex]$193 \pm 2.764\left(\frac{62.73}{\sqrt{10}}\right)$[/tex]

C. [tex]$193 \pm 2.821\left(\frac{62.73}{\sqrt{9}}\right)$[/tex]

D. [tex]$193 \pm 2.821\left(\frac{62.73}{\sqrt{10}}\right)$[/tex]

To find the correct 98% confidence interval for the mean difference in scores (after - before), follow these steps:

1. Identify the Mean and Standard Deviation:
- The mean of the differences is 193 points.
- The standard deviation of the differences is 62.73 points.

2. Determine the Sample Size:
- There are 8 students, so the sample size is 8.

3. Calculate the Standard Error:
- The standard error is calculated by dividing the standard deviation by the square root of the sample size.
- Standard Error = [tex]$ \frac{62.73}{\sqrt{8}} $[/tex].

4. Find the t-value:
- For a 98% confidence level and degrees of freedom (df) equal to the sample size minus one (8 - 1 = 7), the t-value is 2.821. This is found using a t-table.

5. Calculate the Margin of Error:
- The margin of error is found by multiplying the t-value by the standard error.
- Margin of Error = [tex]$ 2.821 \times \text{Standard Error} $[/tex].

6. Determine the Confidence Interval:
- Subtract the margin of error from the mean difference to find the lower limit of the confidence interval.
- Add the margin of error to the mean difference to find the upper limit of the confidence interval.
- So, the confidence interval is given by:
- Lower Limit = 193 - Margin of Error
- Upper Limit = 193 + Margin of Error

Following these steps, the 98% confidence interval for the mean difference in scores is approximately (130.43, 255.57) points.