Question

A simple random sample with

n = 56
provided a sample mean of 29.5 and a sample standard deviation of 4.4.
A. Develop a 90% confidence interval for the population mean.
B. Develop a 95% confidence interval for the population mean.
C. Develop a 99% confidence interval for the population mean.

Answer 1

A simple random sample with n = 56 provided a sample mean of 29.5 and a sample standard deviation of 4.4. The 90%, 95%, and 99% confidence intervals for the population mean are (28.24, 30.76), (27.98, 31.02), and (27.22, 31.78) respectively.

Step-by-step explanation:

A. To develop a 90% confidence interval for the population mean, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation / √(Sample Size))

Plugging in the given values, we have:

Confidence Interval = 29.5 ± (z) * (4.4 / √(56))

Using a z-score of 1.645 (this corresponds to a one-tailed test at 90% confidence level), we can calculate the lower and upper bounds of the confidence interval:

Lower bound = 29.5 - (1.645) * (4.4 / √(56))

Upper bound = 29.5 + (1.645) * (4.4 / √(56))

Therefore, the 90% confidence interval for the population mean is (28.24, 30.76).

B. To develop a 95% confidence interval for the population mean, we follow the same steps as in part A, but use a z-score of 1.96 (corresponds to a one-tailed test at 95% confidence level):

Lower bound = 29.5 - (1.96) * (4.4 / √(56))

Upper bound = 29.5 + (1.96) * (4.4 / √(56))

Therefore, the 95% confidence interval for the population mean is (27.98, 31.02).

C. To develop a 99% confidence interval for the population mean, we use a z-score of 2.576 (corresponds to a one-tailed test at 99% confidence level):

Lower bound = 29.5 - (2.576) * (4.4 / √(56))

Upper bound = 29.5 + (2.576) * (4.4 / √(56))

Therefore, the 99% confidence interval for the population mean is (27.22, 31.78).