Question

Construct the indicated confidence interval for the difference between the two population means.
Assume that the two samples are independent simple random samples selected from normally
distributed populations. Do not assume that the population standard deviations are equal. A paint
manufacturer wished to compare the drying times of two different types of paint. Independent
simple random samples of 11 cans of type A and 9 cans of type B were selected and applied to
similar surfaces. The drying times, in hours, were recorded. The summary statistics are as follows.

Type A

Xba = 75.7 hrs.

s1 = 4.5 hrs.

n1 = 11

Type B

Xba = 64.3 hrs.

s2 = 5.1 hrs.

n2 = 9

Construct a 98% confidence interval for μ1 - μ2, the difference between the mean drying time for
paint of type A and the mean drying time for paint of type B.
4)
A) 6.08 hrs < μ1 - μ2 < 16.72 hrs B) 5.85 hrs < μ1 - μ2 < 16.95 hrs
C) 5.92 hrs < μ1 - μ2 < 16.88 hrs D) 5.78 hrs < μ1 - μ2 < 17.02 hrs

Answer 1

We need to construct the 98% 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	75.7
Sample Standard Deviation 1	4.5
Sample Size 1	11
Sample Mean 2	64.3
Sample Standard Deviation 2	5.1
Sample Size 2	9

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 = 11 + 9 - 2 = 18

The critical value for α=0.02 and df = 18 degrees of freedom is t_c = 2.552. 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 = (5.92, 16.88) ..

C) 5.92 hrs < μ1 - μ2 < 16.88 hrs