STAT 350

Postscript version of these questions

Assignment 6

1.

Consider a design with 5 data points x₁=-2, x₂ = -1, x₃ =0, x₄=1 and x₅=2 so that the design matrix is

$$\left[ \begin{array}{rrr} 1 & -2 \\ 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{array}\right]$$

If we fit a simple linear regression of the form

$$Y_i = \beta_0 + \beta_1 x_i + \epsilon_i$$

for which the design matrix is as above, evaluate the non-centrality parameter of the t test of $\beta_1=0$ when in fact $\beta_1=2$ and $\sigma^2=2.5$. What would the power of a two sided 1% level t test of this null hypothesis be? How many times would we have to replicate this design to get a power of 0.9 for a 1% level test?

2.

For the design points x_i as in the previous question evaluate the non-centrality paramter of the F test of the hypothesis $\beta_1=\beta_2=0$ in the model

$$Y_i = \beta_0 + \beta_1 x_i + \beta_2(x_i^2-2)+\epsilon_i\, .$$

Assume that in fact $\beta_1=1$ and $\beta_2 =2$.

3.

Question 10.7 parts a, c, d, e and f.

4.

Suppose

$$Y_i = \beta_0 + \epsilon_i$$

where the errors $\epsilon_i$ have variances $\sigma^2/w_i$ and the w_i are known quantities. Find an explicit algebraic formula for the weighted least squares estimate of $\beta_0$.