Question

A simple random sample with n = 58 provided a sample mean of 23.5 and a sample standard deviation of 4.4. (Round your answers to one decimal place.) (a) Develop a 90% confidence interval for the population mean. to (b) Develop a 95% confidence interval for the population mean. to (C) Develop a 99% confidence interval for the population mean. to (d) What happens to the margin of error and the confidence interval as the confidence level is increased? As the confidence level increases, there is a smaller margin of error and a more narrow confidence interval. As the confidence level increases, there is a larger margin of error and a wider confidence interval. As the confidence level increases, there is a larger margin of error and a more narrow confidence interval. As the confidence level increases, there is a smaller margin of error and a wider confidence interval.

Answer 1

a) 90% Confidence Interval: (22.4, 24.6). b) 95% Confidence Interval: (22.1, 24.9). c) 99% Confidence Interval: (21.7, 25.3). d) As confidence level increases, margin of error and confidence interval increase.

Step-by-step explanation:

To develop a confidence interval for the population mean, we use the formula:

Confidence Interval = Sample Mean ± (Critical Value)(Standard Error)

(a) For a 90% confidence level:

Sample Mean = 23.5

Sample Standard Deviation = 4.4

Critical Value = 1.645 (from a t-distribution with n-1 degrees of freedom)

Plugging in the values:

Lower Bound = 23.5 - (1.645)(4.4/√58) = 22.381

Upper Bound = 23.5 + (1.645)(4.4/√58) = 24.619

The 90% confidence interval for the population mean is (22.4, 24.6).

(b) For a 95% confidence level:

Sample Mean = 23.5

Sample Standard Deviation = 4.4

Critical Value = 1.96 (from a t-distribution with n-1 degrees of freedom)

Plugging in the values:

Lower Bound = 23.5 - (1.96)(4.4/√58) = 22.146

Upper Bound = 23.5 + (1.96)(4.4/√58) = 24.854

The 95% confidence interval for the population mean is (22.1, 24.9).

(c) For a 99% confidence level:

Sample Mean = 23.5

Sample Standard Deviation = 4.4

Critical Value = 2.626 (from a t-distribution with n-1 degrees of freedom)

Plugging in the values:

Lower Bound = 23.5 - (2.626)(4.4/√58) = 21.709

Upper Bound = 23.5 + (2.626)(4.4/√58) = 25.291

The 99% confidence interval for the population mean is (21.7, 25.3).

(d) As the confidence level increases, the margin of error and the confidence interval both increase. This is because a higher confidence level requires a larger critical value, which in turn increases the margin of error and widens the confidence interval.