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 detemined the mean age to be 37 years with a standard deviation s

1
​
=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 s
2
​
=6.9 years. Let μ
1
​
and μ
2
​
represent the mean ages of the populations of all registered Republicans and Democrats, respectively. Find the 99% confidence interval for the difference μ
1
​
−μ
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 between the mean ages of registered Republicans and Democrats, we can use the formula: Confidence Interval = (X1 - X2) ± Z * √((S1^2/n1) + (S2^2/n2)). Plugging in the given values gives a confidence interval of approximately [-0.978, 2.978].

Step-by-step explanation:

To find the 99% confidence interval for the difference between the mean ages of registered Republicans and Democrats, we can use the formula:

Confidence Interval = (X1 - X2) ± Z * √((S1^2/n1) + (S2^2/n2))

where X1 and X2 are the sample means, S1 and S2 are the sample standard deviations, n1 and n2 are the sample sizes, and Z is the critical value for a 99% confidence level.

Plugging in the given values, we have:

Confidence Interval = (37 - 36) ± 2.576 * √((25/132) + (47.61/120))

Simplifying the calculation gives:

Confidence Interval = (37 - 36) ± 2.576 * √(0.189 + 0.397)

Confidence Interval ≈ 1 ± 2.576 * √(0.586)

Confidence Interval ≈ 1 ± 2.576 * 0.766

Confidence Interval ≈ 1 ± 1.978

Confidence Interval ≈ [-0.978, 2.978]