Suppose that we have Y 1 ,, Y m N( 1 ,), Y m+ 1 ,

Suppose that we have Y₁,…, Y_m ∼ N(μ₁,σ), Y_m+1, …, Y_m+n ∼ N(μ₂, σ), and all m + n observations are independent. These are the assumptions of the pooled t procedure in Section 10.2. Let k = 1, x₁₁ = .5, …, x_m1 = .5, x_m+1,1 = −.5, …, x_m+n,1 = −.5. For convenience in inverting X′X assume m = n.

a. Obtain β̂₀ and β̂₁ from Equation (12.16). Let y̅₁ be the mean of the first m observations and y̅₂ be the mean of the next n observations.

Equation (12.16)

b. Find simple expressions for ŷ, SSE, s_e,  and  s_β̂1.
c. Use parts (a) and (b) to find a simple expression for the 95% CI for β̂₁. Show that your formula is equivalent to

which is the pooled variance confidence interval discussed in Section 9.2.

d.

Transcribed Image Text:

B = b = (X'X)-¹X'y