Construct the indicated confidence interval for the differen

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. Also assume that the population standard deviations are equal 1=2, so that the standard error of the difference between means is obtained by pooling the sample variances .
A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs.

    
Construct a 95% confidence interval for the difference between the mean GPA of college A students and the mean GPA of college B students.
(Note: x1 = 3.1125, x2 = 3.4385, s1 = 0.4357, s2 = 0.5485.)

Solution

We define the difference in mean GPA to be ud = u1 - u2.

So,

xd = x1 - x2 = -0.32596

Also, pooling to get the standard deviation,

s = 0.503795

So, the standard error is

sd = s / sqrt(n1 + n2) = 0.1141

The df = n1 + n2 - 2 = 19. Thus, the critical t is

tcrit = 2.09302

Thus, the margin of error is

E = sd tcrit = 0.2388

Thus,

upper bound = -0.32596 + 0.2388 = -0.0871
lower bound = -0.32596 - 0.2388 = -0.5648 [ANSWER is (-0.5648, -0.0871)]