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Answer :
To construct and interpret a 90% confidence interval for the mean endowment of all private colleges in the United States, follow these steps:
i. Assumptions:
- Sample size is greater than 30. This assumption is satisfied because the sample size is 47.
- Simple random sample: Assumed based on context (not directly stated).
- Population standard deviation is unknown, hence we use the sample standard deviation.
ii. Unknown parameter:
- The unknown parameter we want to estimate is the population mean of endowments.
iii. Point estimate:
- The point estimate of the population mean is the sample mean, which is [tex]$175.1$[/tex] million dollars.
iv. Confidence level and alpha:
- The confidence level provided is 90%.
- Alpha ([tex]\(\alpha\)[/tex]) is calculated as [tex]\(1 - \text{confidence level} = 0.10\)[/tex].
- [tex]\(\frac{\alpha}{2} = 0.05\)[/tex] because this is a two-tailed test.
- The critical value, which is the z-score corresponding to a 90% confidence level and [tex]\(\frac{\alpha}{2} = 0.05\)[/tex], is approximately 1.645 (rounded to 3 decimal places).
v. Margin of error:
- The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = \text{Critical Value} \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}}\right)
\][/tex]
- Using the critical value (1.645), sample standard deviation (131.23 millions), and sample size (47), the margin of error is approximately 31.49 million dollars (rounded to 2 decimal places).
vi. Confidence interval:
- The confidence interval for the population mean is computed by adding and subtracting the margin of error from the sample mean.
- Lower limit: [tex]\(175.1 - 31.49 = 143.61\)[/tex] (rounded to 2 decimal places).
- Upper limit: [tex]\(175.1 + 31.49 = 206.59\)[/tex] (rounded to 2 decimal places).
Therefore, the 90% confidence interval is approximately (143.61, 206.59).
Interpretation:
We are 90% confident that the true mean endowment for all private colleges in the United States falls between [tex]$143.61$[/tex] million and [tex]$206.59$[/tex] million.
i. Assumptions:
- Sample size is greater than 30. This assumption is satisfied because the sample size is 47.
- Simple random sample: Assumed based on context (not directly stated).
- Population standard deviation is unknown, hence we use the sample standard deviation.
ii. Unknown parameter:
- The unknown parameter we want to estimate is the population mean of endowments.
iii. Point estimate:
- The point estimate of the population mean is the sample mean, which is [tex]$175.1$[/tex] million dollars.
iv. Confidence level and alpha:
- The confidence level provided is 90%.
- Alpha ([tex]\(\alpha\)[/tex]) is calculated as [tex]\(1 - \text{confidence level} = 0.10\)[/tex].
- [tex]\(\frac{\alpha}{2} = 0.05\)[/tex] because this is a two-tailed test.
- The critical value, which is the z-score corresponding to a 90% confidence level and [tex]\(\frac{\alpha}{2} = 0.05\)[/tex], is approximately 1.645 (rounded to 3 decimal places).
v. Margin of error:
- The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = \text{Critical Value} \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}}\right)
\][/tex]
- Using the critical value (1.645), sample standard deviation (131.23 millions), and sample size (47), the margin of error is approximately 31.49 million dollars (rounded to 2 decimal places).
vi. Confidence interval:
- The confidence interval for the population mean is computed by adding and subtracting the margin of error from the sample mean.
- Lower limit: [tex]\(175.1 - 31.49 = 143.61\)[/tex] (rounded to 2 decimal places).
- Upper limit: [tex]\(175.1 + 31.49 = 206.59\)[/tex] (rounded to 2 decimal places).
Therefore, the 90% confidence interval is approximately (143.61, 206.59).
Interpretation:
We are 90% confident that the true mean endowment for all private colleges in the United States falls between [tex]$143.61$[/tex] million and [tex]$206.59$[/tex] million.
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