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Answer :
To find the correct 98% confidence interval for the mean difference in scores (after - before), we need to calculate the margin of error and then use it to construct the confidence interval around the given mean difference.
Here's a step-by-step process:
1. Mean Difference: The mean of the differences is given as 193 points.
2. Standard Deviation of Differences: The standard deviation provided is 62.73 points.
3. Determine the Appropriate t-value: Since the confidence level is 98%, and the degree of freedom matches one of the given t-values, we use the appropriate t-value given as either 2.821 (with a sample size treated as 9) or 2.764 (with a sample size treated as 10).
4. Sample Size: The sample size should match the degree of freedom for the t-value. Let's check both options:
- Sample Size = 9: Uses t-value of 2.821
- Sample Size = 10: Uses t-value of 2.764
5. Calculate the Margin of Error for Each Option:
- For the t-value 2.821 with sample size 9:
[tex]\[
\text{Margin of Error} = 2.821 \left( \frac{62.73}{\sqrt{9}} \right)
\][/tex]
- Margin of Error = 58.98711
- For the t-value 2.764 with sample size 10:
[tex]\[
\text{Margin of Error} = 2.764 \left( \frac{62.73}{\sqrt{10}} \right)
\][/tex]
- Margin of Error = 54.82938
6. Construct the Confidence Intervals:
- If using the t-value 2.821 (sample size 9):
- Confidence Interval: [tex]\( 193 \pm 58.98711 \)[/tex]
- Resulting Interval: (134.01289, 251.98711)
- If using t-value 2.764 (sample size 10):
- Confidence Interval: [tex]\( 193 \pm 54.82938 \)[/tex]
- Resulting Interval: (138.17062, 247.82938)
From this detailed evaluation, the correct confidence interval for the mean difference, taking into account the appropriate t-value and degrees of freedom, is the interval from (134.01289 to 251.98711), which corresponds to option [tex]\(193 \pm 2.821\left(\frac{62.73}{\sqrt{9}}\right)\)[/tex].
Here's a step-by-step process:
1. Mean Difference: The mean of the differences is given as 193 points.
2. Standard Deviation of Differences: The standard deviation provided is 62.73 points.
3. Determine the Appropriate t-value: Since the confidence level is 98%, and the degree of freedom matches one of the given t-values, we use the appropriate t-value given as either 2.821 (with a sample size treated as 9) or 2.764 (with a sample size treated as 10).
4. Sample Size: The sample size should match the degree of freedom for the t-value. Let's check both options:
- Sample Size = 9: Uses t-value of 2.821
- Sample Size = 10: Uses t-value of 2.764
5. Calculate the Margin of Error for Each Option:
- For the t-value 2.821 with sample size 9:
[tex]\[
\text{Margin of Error} = 2.821 \left( \frac{62.73}{\sqrt{9}} \right)
\][/tex]
- Margin of Error = 58.98711
- For the t-value 2.764 with sample size 10:
[tex]\[
\text{Margin of Error} = 2.764 \left( \frac{62.73}{\sqrt{10}} \right)
\][/tex]
- Margin of Error = 54.82938
6. Construct the Confidence Intervals:
- If using the t-value 2.821 (sample size 9):
- Confidence Interval: [tex]\( 193 \pm 58.98711 \)[/tex]
- Resulting Interval: (134.01289, 251.98711)
- If using t-value 2.764 (sample size 10):
- Confidence Interval: [tex]\( 193 \pm 54.82938 \)[/tex]
- Resulting Interval: (138.17062, 247.82938)
From this detailed evaluation, the correct confidence interval for the mean difference, taking into account the appropriate t-value and degrees of freedom, is the interval from (134.01289 to 251.98711), which corresponds to option [tex]\(193 \pm 2.821\left(\frac{62.73}{\sqrt{9}}\right)\)[/tex].
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