STAT3600

Linear Statistical Analysis

1.   [49] Consider the data of five observations.

a.   [5]  Write  down the simple linear regression model of yi  on xi .  What are the four model assumptions? State them clearly.

b.   [5] Based on your answers to part (a), find the mean and variance ofSxy. Hint:

c.   [5] Find the least squares estimates of the parameters. Interpret the estimate for the population slope.

d.   [15] Construct the following ANOVA table by filling in the blanks led by letters from A to I.

At 5% significance level, test whether there is a linear relationship between the independent and dependent variables using the information on the ANOVA table. State clearly the null and alternative hypotheses, test statistic, null distribution, decision rule and conclusion.

e.   [3] Construct a 95% confidence interval for the slope.

f.    [4]  Find the coefficient of determination and interpret the result. What is the sample correlation between the independent and dependent variables?

g.   [2] Find a point estimate for the population mean of Y when x is 35.

h.   [4] Construct a 90% confidence interval for the population mean of Y when x is 35.

i.    [6] Predict the individual response (Y) when x is 27. And construct a 90% confidence interval for the individual response.

2.   [51] You are given the following matrices computed from a multiple linear regression of

yi  = β0 + β1xi1 + β2xi2 + εi:

The matrices are properly ordered according to the regression equation given above.

a.   [4] Find the sample size and the sample mean of Y.

b.   [5] It is known that the least square estimator for β is β(^) = (XTX)一1XTY. Using matrix

algebra and linear model assumptions, show that E(β(^)) = β and Var(β(^)) = σ2 (XTX)一1.

c.   [5] Find the least squares estimates for β0, β1 and β2. Interpret the estimates for β1 and β2.

d.   [15] Construct the ANOVA table and hence, test whether the coefficients for the independent variables are jointly equal to zero at the 5% level of significance. Clearly define the null and alternative hypotheses and decision rule. And state your conclusion.

e.   [4] Without proof, write down an unbiased estimator for σ2. Produce an estimate for σ 2  using this estimator.

f.    [9] Testβ2 = 0 at the 5% level of significance. State clearly the null and alternative hypotheses, test statistic, null distribution, decision rule and conclusion.

g.   [9] Test β 1  + β2  =  1 at the  5% level of significance.  State clearly the null and alternative hypotheses, test statistic, null distribution, decision rule and conclusion.