STAT3600
Linear Statistical Analysis
1. [49] Consider the data of five observations.
|
i
|
xi
|
yi
|
|
1
|
46
|
3.5
|
|
2
|
20
|
1.9
|
|
3
|
52
|
4.0
|
|
4
|
30
|
2.6
|
|
5
|
57
|
4.5
|
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.
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Source
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SS
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df
|
MS
|
|
SSR
|
A
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D
|
H
|
|
SSE
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B
|
E
|
I
|
|
SST
|
C
|
F
|
|
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.