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 $132.00. assume the population standard deviation is $19.50. construct a 90% confidence interval for the population mean. the 90% confidence interval is (126.93, 137.07). (round to two decimal places as needed.) construct a 95% confidence interval for the population mean. the 95% confidence interval is (125.96, 138.04). (round to two decimal places as needed.) interpret the results. choose the correct answer below. a. with 90% confidence, it can be said that the population mean price lies in the first interval. with 95% confidence, it can be said that the population mean price lies in the second interval. the 95% confidence interval is wider than the 90%. b. with 90% confidence, it can be said that the population mean price lies in the first interval. with 95% confidence, it can be said that the population mean price lies in the second interval. the 95% confidence interval is narrower than the 90%. c. with 90% confidence, it can be said that the sample mean price lies in the first interval. with 95% confidence, it can be said that the sample mean price lies in the second interval. the 95% confidence interval is wider than the 90%.

Explanation:

• Confidence intervals estimate a range for the population parameter (here, the population mean price). A 90% confidence interval means that if we were to take many samples, about 90% of the intervals constructed would contain the true population mean. Similarly, for a 95% confidence interval, about 95% of such intervals would contain the true population mean.
• The width of a confidence interval is related to the level of confidence. A higher confidence level (like 95% compared to 90%) requires a wider interval. This is because to be more confident that the interval contains the true population mean, we need to cover a larger range of possible values.

Answer:

A. With 90% confidence, it can be said that the population mean price lies in the first interval. With 95% confidence, it can be said that the population mean price lies in the second interval. The 95% confidence interval is wider than the 90%.