Question

Assume that you have a sample of n 1 equals 8​, with the sample mean Xbar1 equals 50​, and a sample standard deviation of S1 equals 7​, and you have an independent sample of n2 equals 15 from another population with a sample mean of Xbar2 equals 33​, and the sample standard deviation S2 equals 5. Construct a 95​% confidence interval estimate of the population mean difference between mu 1 and mu 2. Assume that the two population variances are equal. (blank) is less than or equal to mu1 minus mu2 which is less than (blank). What is the 95% confidence interval

Answer 1

We need to construct the 95% 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	50
Sample Standard Deviation 1	7
Sample Size 1	8
Sample Mean 2	33
Sample Standard Deviation 2	5
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 = 8 + 15 - 2 = 21

The critical value for α=0.05 and df = 21 degrees of freedom is t_c = 2.08 . 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:

CI = (11.77, 22.23)