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Answer :
To determine the 98% confidence interval for the mean difference in scores (after - before), follow these steps:
1. Calculate the Mean Difference:
The mean difference given is 193 points.
2. Find the Standard Deviation:
The standard deviation of the differences is 62.73 points.
3. Determine the Sample Size:
The sample size is 8 students.
4. Degrees of Freedom:
For the t-distribution, the degrees of freedom is the sample size minus one. Thus, degrees of freedom = 8 - 1 = 7.
5. Locate the t Value:
For a 98% confidence level and 7 degrees of freedom, you use a t-distribution table to find the t value. The correct t* value provided is 2.821.
6. Calculate the Standard Error:
The standard error (SE) is found using the formula:
[tex]\[
SE = \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}}
\][/tex]
So:
[tex]\[
SE = \frac{62.73}{\sqrt{8}} \approx 22.18
\][/tex]
7. Compute the Margin of Error:
The margin of error (ME) is calculated using:
[tex]\[
ME = t^* \times SE
\][/tex]
So:
[tex]\[
ME = 2.821 \times 22.18 \approx 62.57
\][/tex]
8. Determine the Confidence Interval:
The 98% confidence interval is the mean difference plus or minus the margin of error:
- Lower Bound = 193 - 62.57 ≈ 130.43
- Upper Bound = 193 + 62.57 ≈ 255.57
Therefore, the 98% confidence interval for the mean difference in scores is approximately (130.43, 255.57).
1. Calculate the Mean Difference:
The mean difference given is 193 points.
2. Find the Standard Deviation:
The standard deviation of the differences is 62.73 points.
3. Determine the Sample Size:
The sample size is 8 students.
4. Degrees of Freedom:
For the t-distribution, the degrees of freedom is the sample size minus one. Thus, degrees of freedom = 8 - 1 = 7.
5. Locate the t Value:
For a 98% confidence level and 7 degrees of freedom, you use a t-distribution table to find the t value. The correct t* value provided is 2.821.
6. Calculate the Standard Error:
The standard error (SE) is found using the formula:
[tex]\[
SE = \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}}
\][/tex]
So:
[tex]\[
SE = \frac{62.73}{\sqrt{8}} \approx 22.18
\][/tex]
7. Compute the Margin of Error:
The margin of error (ME) is calculated using:
[tex]\[
ME = t^* \times SE
\][/tex]
So:
[tex]\[
ME = 2.821 \times 22.18 \approx 62.57
\][/tex]
8. Determine the Confidence Interval:
The 98% confidence interval is the mean difference plus or minus the margin of error:
- Lower Bound = 193 - 62.57 ≈ 130.43
- Upper Bound = 193 + 62.57 ≈ 255.57
Therefore, the 98% confidence interval for the mean difference in scores is approximately (130.43, 255.57).
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