Suppose X₁,...,X_m is an independent random sample of size m from a population which is normally distributed with mean μ₁ and variance σ²₁. Let Y₁,...Y_n be another independent random sample of size n from a population which is normally distributed wieth mean μ₂ ad variance σ²₂. Let X (with bar) and Y (with bar) denote the sample means from these two samples, respectively since the two samples are independent of each other, the corresponding sample means are also independent.

a. What are the sampling distributiongs of X (with bar over it) and Y (with bar)? Give their names and parameters

b. It is known that, both, the sum and and the difference of two independent random variables that follow normal distributions are also normally distributed random variables. Using this information, what are the distributions of X(with bar) + Y (with bar) and X (with bar) - Y(with bar)? Give their names and parameters.

c. Let σ²₁ = 4 and σ²₂ = 1, and m = 32 and n = 8. Then, find the probability that X(with bar) - Y(with bar) is within 1 unit of the difference in true populaiton means μ₁ - μ₂