8.2 Let X1,...,Xn a random sample from an N(, 2) distribution. a) Show that the random variable X /n is distributed as a

8.2 Let X1,...,Xn a random sample from an N(μ, σ2) distribution.

a) Show that the random variable X¯ − μ

σ/√n is distributed as a N(0, 1).

b) Show that the random variable 1

σ2 S2 = 1

σ2 1

n − 1

n i=1

(Xi − X¯ )

2 has the same distribution as 1 n−1χ2 n−1, where χ2 n−1 is a Chi-squared random variable with n − 1 degrees of freedom.