Suppose 1, , is an independent random sample of size m from a population which is normally distributed with mean 1 and variance 1 2 Let 1 , , n be another independent random sample of size n from a population which is normally distributed with mean 2 and variance 2 2 Let (bar) and (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 distributions of and Give their names and parameters (5 pts) (b)It is known that, both, the sum 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 (bar) (bar)and (bar) (bar) Give their names and parameters Hint You just need to find the mean and variance of (bar) (bar) and (bar) (bar) Note, if X and Y are two independent random variables, then V(X Y) V(X) V(Y), and we know from property of variance that V( Y) V(Y) (8 pts) (c) Let 1 2 4 and 2 2 1, and m 32 and n 8 Then, find the probability that (bar) (bar) is within 1 unit of the difference in true population means 1 2 (7 pts)

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Suppose 1, ... , is an independent random sample of size m from a population which is normally distributed with mean 1 and variance 1^2.

Suppose 1, ... , is an independent random sample of size m from a population which is normally distributed with mean 1 and variance 1^2. Let 1 , ... , n be

another independent random sample of size n from a population which is normally distributed with mean 2 and variance 2^2 . Let (bar) and (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 distributions of and ? Give their names and parameters. (5 pts)
(b)It is known that, both, the sum 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 (bar)+ (bar)and (bar)-(bar)? Give their names and parameters. [Hint: You just need to find the mean and variance of (bar)+ (bar) and (bar)-(bar) . Note, if X and Y are two independent random variables, then V(X+Y)=V(X)+V(Y), and we know from property of variance that V(-Y)=V(Y)] (8 pts)

(c) Let 1^2 =4 and 2^2 =1, and m=32 and n=8. Then, find the probability that (bar)-(bar) is within 1 unit of the difference in true population means 1 2. (7 pts)