Question

An economist wants to compare the average duration of unemployment in two counties. A simple random sample of 12 unemployed persons in Mountain County shows an average duration of 37.2 weeks, with a standard deviation of 12.7 weeks. A simple random sample of 15 unemployed persons in Valley County reveals an average duration of 34.1 weeks, with a standard deviation of 2.3 weeks. Construct a 99 percent confidence interval for the difference between the population means.

Answer 1

We need to construct the 99% confidence interval for the difference between the population means μ1​−μ2​, for the case that the population standard deviations are not known. The following information has been provided about each of the samples:

Sample Mean 1	37.2
Sample Standard Deviation 1	12.7
Sample Size 1	12
Sample Mean 2	34.1
Sample Standard Deviation 2	2.3
Sample Size 2	15

Based on the information provided, we assume that the population variances are equal, so then the number of degrees of freedom are df = n_1 + n_2 -2 = 12 + 15 - 2 = 25

The critical value for \alpha = 0.01α=0.01 and df = 25 degrees of freedom is t_c = 2.787. The corresponding confidence interval is computed as shown below:

Since the population variances are assumed to be equal, we need to compute the pooled standard deviation, as follows:

Since we assume that the population variances are equal, the standard error is computed as follows:

Now, we finally compute the confidence interval: