Math Statistics Question - Sampling Distribution

Independent random samples of 64 observations each are chosen from two normal populations with the following means and standard deviations:
Population 1 Population 2
µ1 = 12 µ2 = 10
?1 = 4 ?2 = 3
Let x-bar1 and x-bar2 denote the two sample means.
a. Give the mean and standard deviation of the sampling distribution of x-bar 1 .

b. Give the mean and standard deviation of the sampling distribution of x-bar2 .

c. Suppose you were to calculate the difference (x-bar1 - x-bar2) between the sample means. Find the mean and standard deviation of the sampling distribution of (x-bar1 - x-bar2).

d. Will the statistic (x-bar1 - x-bar2) be normally distributed? Explain.

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n1=n2=64; µ1 = 12, µ2 = 10; σ1=4, σ2=3. By theorems,
X-bar 1 ~ N(12, 16/64), standard deviation 4/8
X-bar 2 ~ N(10, 9/64),standard deviation 3/8
(X-bar1 - X-bar2) ~ (12-10, 16/64+9/64), standard deviation = sqrt (16/64+9/64).
Sure, normally distributed. Why, you must have seen the related theorem!