Question

A political analyst was curious if younger adults were becoming more conservative. He decided to see if the mean age of registered Republicans was lower than that of registered Democrats. He selected a simple random sample of 132 registered Republicans from a list of registered Republicans and determined the mean age to be 37 years with a standard deviation s1 = 5 years. He also selected an independent simple random sample of 120 registered Democrats from a list of registered Democrats and determined the mean age to be 36 years with a standard deviation s2 = 6.9 years. Let mu 1 and mu 2 represent the mean ages of the populations of all registered Republicans and Democrats, respectively.

Find the 99% confidence interval for the difference mu 1 – mu 2 (please be sure to also show the calculation of the margin of error E).

Answer 1

To find the 99% confidence interval for the difference in mean ages between registered Republicans and Democrats, we can calculate the margin of error (E) and use it to construct the confidence interval. The margin of error is calculated using the formula E = Z * sqrt((s1^2/n1) + (s2^2/n2)), where Z is the z-score corresponding to the desired confidence level, s1 and s2 are the standard deviations of the samples, and n1 and n2 are the sample sizes. Once we have the margin of error, we can calculate the confidence interval as (mean1 - mean2) ± E.

Step-by-step explanation:

We are given two samples: a sample of 132 registered Republicans with a mean age of 37 years and a standard deviation of 5 years, and a sample of 120 registered Democrats with a mean age of 36 years and a standard deviation of 6.9 years. We want to find the 99% confidence interval for the difference in mean ages between the two populations.

To calculate the confidence interval, we need to calculate the margin of error (E) first. The formula for the margin of error is:

E = Z * sqrt((s1^2/n1) + (s2^2/n2))

In this formula, Z is the z-score corresponding to the desired confidence level (99% confidence level corresponds to a z-score of approximately 2.576), s1 and s2 are the standard deviations of the two samples, n1 and n2 are the sample sizes.

Plugging in the values from the question, we have:

E = 2.576 * sqrt((5^2/132) + (6.9^2/120))

Calculating this expression gives us E ≈ 0.9837.

The confidence interval is given by: (mean1 - mean2) ± E

Plugging in the values from the question, we have:

Confidence interval = (37 - 36) ± 0.9837

Calculating this expression gives us a confidence interval of approximately (0.0163, 1.9837). Therefore, we can be 99% confident that the true difference in mean ages between registered Republicans and Democrats is between 0.0163 and 1.9837 years.