29. The regression model Y = x + e, e N(0,2) is called regression through the origin since it presupposes that the expected response

29. The regression model Y = βx +

e, e ∼ N(0,σ2)

is called regression through the origin since it presupposes that the expected response corresponding to the input level x=0 is equal to 0.

Suppose that (xi,Yi ), i = 1, . . . , n is a data set from this model.

a. Determine the least squares estimator B of β.

b. What is the distribution of B?

c. Define SSR and give its distribution.

d. Derive a test of H0 : β = β0 versus H1 : β = β0.

e. Determine a 100(1 − α) percent prediction interval for Y(x0), the response at input level x0.