Answer :
To find the correct 98% confidence interval for the mean difference in scores (after - before), we can follow these steps:
1. Identify the Given Values:
- Mean of the differences: 193 points.
- Standard deviation of the differences: 62.73 points.
- Sample size: 8 students (since there are 8 differences given).
2. Determine the Degrees of Freedom:
- Degrees of freedom (df) = sample size - 1 = 8 - 1 = 7.
3. Find the Appropriate [tex]\( t \)[/tex]-value:
- For a 98% confidence level and 7 degrees of freedom, the [tex]\( t \)[/tex]-value is approximately 2.821.
4. Calculate the Standard Error:
- Standard error (SE) is calculated using the formula:
[tex]\[
\text{SE} = \frac{\text{Standard deviation}}{\sqrt{\text{Sample size}}}
\][/tex]
- So the standard error is:
[tex]\[
\text{SE} = \frac{62.73}{\sqrt{8}} \approx 22.18
\][/tex]
5. Calculate the Margin of Error:
- Margin of error (ME) is calculated by multiplying the [tex]\( t \)[/tex]-value by the standard error:
[tex]\[
\text{ME} = t \times \text{SE} = 2.821 \times 22.18 \approx 62.57
\][/tex]
6. Determine the Confidence Interval:
- The confidence interval is calculated by adding and subtracting the margin of error from the mean difference:
- Lower bound = Mean difference - ME = 193 - 62.57 = 130.43
- Upper bound = Mean difference + ME = 193 + 62.57 = 255.57
So, the 98% confidence interval for the mean difference in scores is approximately [tex]\( (130.43, 255.57) \)[/tex].
1. Identify the Given Values:
- Mean of the differences: 193 points.
- Standard deviation of the differences: 62.73 points.
- Sample size: 8 students (since there are 8 differences given).
2. Determine the Degrees of Freedom:
- Degrees of freedom (df) = sample size - 1 = 8 - 1 = 7.
3. Find the Appropriate [tex]\( t \)[/tex]-value:
- For a 98% confidence level and 7 degrees of freedom, the [tex]\( t \)[/tex]-value is approximately 2.821.
4. Calculate the Standard Error:
- Standard error (SE) is calculated using the formula:
[tex]\[
\text{SE} = \frac{\text{Standard deviation}}{\sqrt{\text{Sample size}}}
\][/tex]
- So the standard error is:
[tex]\[
\text{SE} = \frac{62.73}{\sqrt{8}} \approx 22.18
\][/tex]
5. Calculate the Margin of Error:
- Margin of error (ME) is calculated by multiplying the [tex]\( t \)[/tex]-value by the standard error:
[tex]\[
\text{ME} = t \times \text{SE} = 2.821 \times 22.18 \approx 62.57
\][/tex]
6. Determine the Confidence Interval:
- The confidence interval is calculated by adding and subtracting the margin of error from the mean difference:
- Lower bound = Mean difference - ME = 193 - 62.57 = 130.43
- Upper bound = Mean difference + ME = 193 + 62.57 = 255.57
So, the 98% confidence interval for the mean difference in scores is approximately [tex]\( (130.43, 255.57) \)[/tex].
