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Answer :
To find the 98% confidence interval for the mean difference in scores (after - before), we need to follow these steps:
1. Identify the Given Values:
- Mean difference ([tex]\(\bar{x}\)[/tex]): 193 points
- Standard deviation of the differences ([tex]\(s\)[/tex]): 62.73 points
- Sample size ([tex]\(n\)[/tex]): 8 students
- Given [tex]\(t\)[/tex]-value for 98% confidence: 2.764
2. Calculate the Standard Error:
The standard error (SE) of the mean difference is calculated using the formula:
[tex]\[
\text{SE} = \frac{s}{\sqrt{n}}
\][/tex]
Where [tex]\(s\)[/tex] is the standard deviation and [tex]\(n\)[/tex] is the sample size.
3. Calculate the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = t \times \text{SE}
\][/tex]
Where [tex]\(t\)[/tex] is the [tex]\(t\)[/tex]-value and [tex]\(\text{SE}\)[/tex] is the standard error.
4. Construct the Confidence Interval:
The confidence interval is given by:
[tex]\[
\text{CI} = \bar{x} \pm \text{ME}
\][/tex]
Where [tex]\(\bar{x}\)[/tex] is the mean difference.
5. Results:
- Standard Error (SE): Approximately 22.18
- Margin of Error (ME): Approximately 61.30
- Confidence Interval: From approximately 131.70 to 254.30
Thus, the correct 98% confidence interval for the mean difference in scores is approximately [tex]\(131.70\)[/tex] to [tex]\(254.30\)[/tex] points. Among the given choices, it corresponds to:
[tex]\[
193 \pm 2.764\left(\frac{62.73}{\sqrt{8}}\right)
\][/tex]
However, it seems there was a mismatch in the calculated standard error. Considering the calculation method, ensure correct values and use trusted [tex]\(t\)[/tex]-value sources to align results accurately.
1. Identify the Given Values:
- Mean difference ([tex]\(\bar{x}\)[/tex]): 193 points
- Standard deviation of the differences ([tex]\(s\)[/tex]): 62.73 points
- Sample size ([tex]\(n\)[/tex]): 8 students
- Given [tex]\(t\)[/tex]-value for 98% confidence: 2.764
2. Calculate the Standard Error:
The standard error (SE) of the mean difference is calculated using the formula:
[tex]\[
\text{SE} = \frac{s}{\sqrt{n}}
\][/tex]
Where [tex]\(s\)[/tex] is the standard deviation and [tex]\(n\)[/tex] is the sample size.
3. Calculate the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = t \times \text{SE}
\][/tex]
Where [tex]\(t\)[/tex] is the [tex]\(t\)[/tex]-value and [tex]\(\text{SE}\)[/tex] is the standard error.
4. Construct the Confidence Interval:
The confidence interval is given by:
[tex]\[
\text{CI} = \bar{x} \pm \text{ME}
\][/tex]
Where [tex]\(\bar{x}\)[/tex] is the mean difference.
5. Results:
- Standard Error (SE): Approximately 22.18
- Margin of Error (ME): Approximately 61.30
- Confidence Interval: From approximately 131.70 to 254.30
Thus, the correct 98% confidence interval for the mean difference in scores is approximately [tex]\(131.70\)[/tex] to [tex]\(254.30\)[/tex] points. Among the given choices, it corresponds to:
[tex]\[
193 \pm 2.764\left(\frac{62.73}{\sqrt{8}}\right)
\][/tex]
However, it seems there was a mismatch in the calculated standard error. Considering the calculation method, ensure correct values and use trusted [tex]\(t\)[/tex]-value sources to align results accurately.
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