Given the following information about a hypothesis test of the difference between two means based on independent random samples, which one of the following is the correct rejection region at a significance level of .05? Assume that the samples are obtained from normally distributed populations having equal variances.

H0: μA ≤ μB, H1: μA >μB 1 = 12, 2 = 9, s1 = 4, s2=2, n1 = 13, n2 = 10.

A. Reject H0 if Z > 1.96
B. Reject H0 if Z > 1.645
C. Reject H0 if t > 2.08
D. Reject H0 if t > 1.782

I'll give you a hint.

H1 tells you if the test is one-tailed or two-tailed. If H1 shows a specific direction, like > or <, then the test is one-tailed. If the test does not show a specific direction, then the test is two-tailed. Therefore, since H1 shows a specific direction, look at the appropriate table for a one-tailed test at .05 level of significance.

I'll let you take it from here.

To determine the correct rejection region at a significance level of 0.05 for a hypothesis test of the difference between two means, we need to calculate the test statistic and compare it to the critical value.

Given:
H0: μA ≤ μB
H1: μA > μB
1 = 12, 2 = 9, s1 = 4, s2 = 2, n1 = 13, n2 = 10.

First, calculate the pooled standard deviation:
Spooled = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2)/(n1+n2-2))
= sqrt(((13-1)*4^2 + (10-1)*2^2)/(13+10-2))
= sqrt((12*16 + 9*4)/(23))
= sqrt((192 + 36)/(23))
= sqrt(228/23)
≈ 2.639

Next, calculate the test statistic (t-statistic):
t = ((x1 - x2) - 0) / (Spooled * sqrt(1/n1 + 1/n2))
= ((12 - 9) - 0) / (2.639 * sqrt(1/13 + 1/10))
= (3) / (2.639 * sqrt(0.077 + 0.1))
= 3 / (2.639 * sqrt(0.177))
= 3 / (2.639 * 0.421)
≈ 3 / 1.111
≈ 2.7

Finally, compare the test statistic to the critical value at a significance level of 0.05. Since we are performing a one-tailed test and the alternative hypothesis is μA > μB, we need to find the critical value in the upper tail.

Looking up the critical value for t-distribution with degrees of freedom (df) = n1 + n2 - 2 = 13 + 10 - 2 = 21 and a significance level of 0.05, we find that t > 1.724.

Therefore, the correct rejection region is:
D. Reject H0 if t > 1.782.

Note: The option values in the answer choices provided do not match the calculated values, so none of the given options is correct in this case.