Let X1 to be normally distributed with mu1 and o^2 1 be the random variable denoting the first normal population and
X2 also normally distributed with mu2 and o^2 2 the random variable denoting the second normal population. The two populations (random variables) X1 and X2 are independent. If X1 is the sample mean in random samples of size n1 from the first population, and X2 is the sample mean in random samples of size n2 from the second population, then X1 and X2 are independent random variables. The difference of the two-sample means, X1 - X2, is also a random variable. Its probability distribution is called the sampling distribution of the difference between the two-sample means.

(a) Using the above information, and your knowledge on sampling distributions and linear combinations of independent random variables, find what is the sampling distribution of the difference between the two-sample means, X1 - X2. What is the expected value (mean) of the difference, E(X1 - Xz), and what is its variance, V (X1 - X2)?

(b) By applying the new knowledge that you acquire on part a), answer the following question:
A population random variable X1 has a normal distribution with mean M1 = 29.8 and
standard deviation 01 = 4. Another population random variable X2 has a normal
distribution with mean M2 = 34.7 and standard deviation 02 = 5. X1 and X2 are independent.
Random samples of size n1 = 20 from the first population X1 and samples of size n2 = 20
from the second population X2 are selected. Denote with X1 - X2 the difference between the two-sample means. State what is the sampling distribution of X4 - X2 and calculate its expected value (mean), E (X1 - X2), and the variance, V (X1 - X2).