STAT3600 Linear Statistical Analysis
1. [49] Consider the data of five observations.
|
i
|
xi
|
yi
|
|
1
|
26
|
3.2
|
|
2
|
23
|
1.8
|
|
3
|
62
|
4.0
|
|
4
|
20
|
2.3
|
|
5
|
17
|
4.8
|
a. [5] Write down the simple linear regression model of yi on xi . What are the four model assumptions? State them clearly.
b. [5] Letβ(^)1 be the least squares estimator for the unknown population slope in the simple linear
regression model. Prove that
c. [5] Find the least squares estimates of the population intercept and slope. 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
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MS
|
|
SSR
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A
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D
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G
|
|
SSE
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B
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E
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H
|
|
SST
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C
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F
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|
e. [6] Using the Bonferroni's method, construct simultaneous confidence intervals for the
population intercept and slope with a family confidence level of at least 95%.
f. [2] Find the coefficient of determination and interpret the result.
g. [1] Find a point estimate for the population mean of Y when x is 25.
h. [4] Construct a 90% confidence interval for the population mean of Y when x is 25.
i. [6] Let Y(1) and Y(2) be future responses with the values of x being 30 and 35, respectively. Construct a 95% prediction interval for Y(1) − Y(2) .
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 r.
b. [5] Show that the least squares estimator for β is given by β(^) = (XTX)-1XTY.
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. State your conclusion.
e. [7] At the 5% level of significance, conduct a t-test for H0 : β1 = β2 vs. H1 : β1 ≠ β2.
f. [6] Construct a 95% confidence interval for β1 + 2β2.
g. [9] Define At the 5% level of significance, test the following hypotheses.
H0 : Cβ = d vs. H1 : Cβ ≠ d.