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To estimate the true mean GPA of the freshman class with 99% confidence, we need to follow several steps involving the sample data, confidence interval computation, and statistical principles. Here is a detailed, step-by-step solution:
1. Calculate the Sample Mean ( [tex]\(\bar{x}\)[/tex] ):
The sample mean is the average of all provided GPA values. Given the GPAs:
[tex]\[ 1.9, 3.2, 2.0, 2.9, 2.7, 3.3, 2.8, 3.0, 3.8, 2.7, 2.0, 1.9, 2.5, 2.7, 2.8, 3.2, 3.0, 3.8, 3.1, 2.7, 3.5, 3.8, 3.9, 2.7, 2.0, 2.8, 1.9, 4.0, 2.2, 2.8, 2.1, 2.4, 3.0, 3.4, 2.9, 2.1 \][/tex]
The calculated sample mean is:
[tex]\[ \bar{x} \approx 2.819 \][/tex]
2. Determine the Sample Size ( [tex]\(n\)[/tex] ):
Count the number of GPA values in the sample. Here,
[tex]\[ n = 36 \][/tex]
3. Standard Error of the Mean ( [tex]\(SE\)[/tex] ):
The standard error of the mean is calculated using the population standard deviation ( [tex]\(\sigma\)[/tex] ) and the sample size ( [tex]\(n\)[/tex] ). The formula is:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
Given [tex]\(\sigma = 0.62\)[/tex],
[tex]\[ SE = \frac{0.62}{\sqrt{36}} = \frac{0.62}{6} \approx 0.103 \][/tex]
4. Z-score for the Given Confidence Level:
For a 99% confidence level, we find the z-score (critical value) corresponding to the desired confidence level. The z-score for a 99% confidence level is approximately:
[tex]\[ z \approx 2.576 \][/tex]
5. Margin of Error ( [tex]\(E\)[/tex] ):
The margin of error is calculated using the z-score and the standard error:
[tex]\[ E = z \times SE \][/tex]
Substituting the values,
[tex]\[ E = 2.576 \times 0.103 \approx 0.266 \][/tex]
6. Confidence Interval:
The confidence interval is found by adding and subtracting the margin of error from the sample mean. Thus, the lower and upper bounds of the confidence interval are:
[tex]\[ \text{Lower bound} = \bar{x} - E \approx 2.819 - 0.266 \approx 2.553 \][/tex]
[tex]\[ \text{Upper bound} = \bar{x} + E \approx 2.819 + 0.266 \approx 3.086 \][/tex]
7. Conclusion:
Based on the calculation, we can estimate the true mean GPA of the freshman class to be in the range:
[tex]\[ (2.553, 3.086) \][/tex]
with 99% confidence.
Therefore, the true mean GPA of the freshman class is estimated to be between 2.553 and 3.086 with a 99% confidence level.
1. Calculate the Sample Mean ( [tex]\(\bar{x}\)[/tex] ):
The sample mean is the average of all provided GPA values. Given the GPAs:
[tex]\[ 1.9, 3.2, 2.0, 2.9, 2.7, 3.3, 2.8, 3.0, 3.8, 2.7, 2.0, 1.9, 2.5, 2.7, 2.8, 3.2, 3.0, 3.8, 3.1, 2.7, 3.5, 3.8, 3.9, 2.7, 2.0, 2.8, 1.9, 4.0, 2.2, 2.8, 2.1, 2.4, 3.0, 3.4, 2.9, 2.1 \][/tex]
The calculated sample mean is:
[tex]\[ \bar{x} \approx 2.819 \][/tex]
2. Determine the Sample Size ( [tex]\(n\)[/tex] ):
Count the number of GPA values in the sample. Here,
[tex]\[ n = 36 \][/tex]
3. Standard Error of the Mean ( [tex]\(SE\)[/tex] ):
The standard error of the mean is calculated using the population standard deviation ( [tex]\(\sigma\)[/tex] ) and the sample size ( [tex]\(n\)[/tex] ). The formula is:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
Given [tex]\(\sigma = 0.62\)[/tex],
[tex]\[ SE = \frac{0.62}{\sqrt{36}} = \frac{0.62}{6} \approx 0.103 \][/tex]
4. Z-score for the Given Confidence Level:
For a 99% confidence level, we find the z-score (critical value) corresponding to the desired confidence level. The z-score for a 99% confidence level is approximately:
[tex]\[ z \approx 2.576 \][/tex]
5. Margin of Error ( [tex]\(E\)[/tex] ):
The margin of error is calculated using the z-score and the standard error:
[tex]\[ E = z \times SE \][/tex]
Substituting the values,
[tex]\[ E = 2.576 \times 0.103 \approx 0.266 \][/tex]
6. Confidence Interval:
The confidence interval is found by adding and subtracting the margin of error from the sample mean. Thus, the lower and upper bounds of the confidence interval are:
[tex]\[ \text{Lower bound} = \bar{x} - E \approx 2.819 - 0.266 \approx 2.553 \][/tex]
[tex]\[ \text{Upper bound} = \bar{x} + E \approx 2.819 + 0.266 \approx 3.086 \][/tex]
7. Conclusion:
Based on the calculation, we can estimate the true mean GPA of the freshman class to be in the range:
[tex]\[ (2.553, 3.086) \][/tex]
with 99% confidence.
Therefore, the true mean GPA of the freshman class is estimated to be between 2.553 and 3.086 with a 99% confidence level.
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