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

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.