Jim works for the housing authority and wants to estimate the mean square footage of apartments for budgeting purposes, so he takes a random sample of 96 apartments. If he knows the population standard deviation of square footage of apartments is 72 space f t squared and his sample has a mean of 605 space f t squared, what would be the endpoints (in f t squared) of a confidence interval for the mean square footage of apartments using a:

a.) 90 percent signconfidence level? The low endpoint is

and the high endpoint is

b.) 99 percent sign confidence level? The low endpoint is

and the high endpoint is

c.) 99.9 percent signconfidence level? The low endpoint is

and the high endpoint is

Round your answers to three decimal places, and do not use units.

To calculate the confidence intervals for the mean square footage of apartments at different confidence levels, we will use the formula for a confidence interval for the population mean:

Confidence Interval = [tex]\bar X \pm Z * (\sigma / \sqrt n)[/tex]

Where:

- [tex]\bar X[/tex] is the sample mean (605)

- Z is the Z-score corresponding to the desired confidence level

- σ is the population standard deviation (72)

- n is the sample size (96)

a) 90% Confidence Level:

Z-score for a 90% confidence level is 1.645

Confidence Interval = 605 ± 1.645 * (72 / √96)

Confidence Interval = 605 ± 1.645 * (72 / 9.798)

Confidence Interval = 605 ± 1.645 * 7.35

Confidence Interval = 605 ± 12.089

Low endpoint: 605 - 12.089 = 592.911

High endpoint: 605 + 12.089 = 617.089

b) 99% Confidence Level:

Z-score for a 99% confidence level is 2.576

Confidence Interval = 605 ± 2.576 * (72 / √96)

Confidence Interval = 605 ± 2.576 * (72 / 9.798)

Confidence Interval = 605 ± 2.576 * 7.35

Confidence Interval = 605 ± 18.97

Low endpoint: 605 - 18.97 = 586.03

High endpoint: 605 + 18.97 = 623.97

c) 99.9% Confidence Level:

Z-score for a 99.9% confidence level is 3.291

Confidence Interval = 605 ± 3.291 * (72 / √96)

Confidence Interval = 605 ± 3.291 * (72 / 9.798)

Confidence Interval = 605 ± 3.291 * 7.35

Confidence Interval = 605 ± 24.19

Low endpoint: 605 - 24.19 = 580.81

High endpoint: 605 + 24.19 = 629.19

Therefore, the confidence intervals for the mean square footage of apartments are:

a) At a 90% confidence level: (592.911, 617.089)

b) At a 99% confidence level: (586.03, 623.97)

c) At a 99.9% confidence level: (580.81, 629.19)