Question

You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals.

A random sample of 40 home theater systems has a mean price of $127.00. Assume the population standard deviation is $16.70.
Construct a 90% confidence interval for the population mean. The 90% confidence interval is ( 122.66 , 131.34 ). (Round to two decimal places as needed.) Construct a 95% confidence interval for the population mean.
The 95% confidence interval is ( 121.82 , 132.16 (Round to two decimal places as needed.) You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals.
From a random sample of 38 business days, the mean closing price of a certain stock was $110.75. Assume the population standard deviation is $11.12.

Answer 1

To construct confidence intervals for population means with a known standard deviation, you need the sample mean, population standard deviation, and desired confidence level. Using the formula (sample mean - error bound, sample mean + error bound), calculate the confidence intervals for different confidence levels. In the given example, the 90% confidence interval is (122.66, 131.34) and the 95% confidence interval is (121.82, 132.16).

Step-by-step explanation:

To construct a confidence interval for a population mean using a known population standard deviation, we need the sample mean, the population standard deviation, and the desired confidence level.

For the 90% confidence interval, we use the formula (sample mean - error bound, sample mean + error bound). The error bound is calculated by multiplying the population standard deviation by the critical value for a 90% confidence level, which can be found in a t-table or using technology.

For the 95% confidence interval, the process is the same, but we use the critical value for a 95% confidence level instead. The 95% confidence interval will end up being wider than the 90% interval because we are more confident in its accuracy.

In the given example, the 90% confidence interval for the population mean of home theater system prices is (122.66, 131.34) and the 95% confidence interval is (121.82, 132.16).