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 35 home theater systems has a mean price of $139.00. Assume the population standard deviation is $19.10.
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Part 1
Construct a 90% confidence interval for the population mean.
The 90% confidence interval is (
enter your response here,
enter your response here).
(Round to two decimal places as needed.)
Part 2
Construct a 95% confidence interval for the population mean.
The 95% confidence interval is (
enter your response here,
enter your response here).
(Round to two decimal places as needed.)
Part 3
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 narrower than the 90%.
B.
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%.
C.
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%.
Part 1: To construct a 90% confidence interval for the population mean, we use the formula: Confidence Interval = sample mean ± Z * (population standard deviation / √sample size) Given: - Sample Mean: $139.00 - Population Standard Deviation: $19.10 - Sample Size: 35 - Z-value for 90% confidence level: 1.645 (from standard normal distribution) Plugging in the values: Confidence Interval = 139.00 ± 1.645 * (19.10 / √35) Calculate the confidence interval.
Part 2: To construct a 95% confidence interval, we use the same formula but with a different Z-value. - Z-value for 95% confidence level: 1.96 Confidence Interval = 139.00 ± 1.96 * (19.10 / √35) Calculate the confidence interval.
Part 3: Compare the widths of the confidence intervals and interpret the results.
