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DEPARTMENT OF STATISTICS ANDACTUARIAL SCIENCE
STAT3600
LINEAR STATISTICALANALYSIS
May 16,2023
1. The data in the following table relate Yand X.
It is given that,
(a) Find and interpret the least squares estimates of the regression coefficients. [6 marks]
(b) Construct the ANOVA table and test whether Xhas any effect on Y based on an F test at the 5%level of significance. State the hypotheses,decision rule and conclusion. [10 marks]
(c) Calculate and interpret the coefficient of determination. [3 marks]
(d) Calculate the sample covariance matrix of the least squares estimates of the regression coefficients. [6 marks]
(e) Estimate the means of Ywhen X=0.05 and -0.05.Find a simultaneous Bonferroni confidence region for the estimation with at least 90%confidence level. [7 marks]
[Total: 32 marks]
2. A regression analysis of Y on X₁ and X₂ with normal errors is considered. Fifty observations are obtained. It is given that SST is 0.205. The values of SSE for various independent variables in a model are given as follows.Conduct a forward selection method with the selection level of an F-value being 3.0.Show the steps of the selection procedure.
[Total:10 marks]
3. A regression analysis of Yon X₁and X₂with normal errors is considered.The following matrices are computed.
The elements of the matrices are properly ordered according to the regression function given above.
(a) Find the least squares estimates of the regression coefficients. [3marks]
(b) Construct an ANOVA table for the regression analysis.Test whether there is a regression between the dependent and the independent variables at the 5%level of significance.State the decision rule and conclusion. [10 marks]
(c) Test the following hypothesis at the 5%level of significance,
H₀:β₁+β₂=0.
State the decision rule and conclusion. [5 marks]
(d) Construct a 95%prediction interval for y₁+2y₂where y₁is a future response
where 
= (0.5, 0.5) and y₂is a future response where 
= (-1,0.5). [6marks]
[Total:24 marks]
4.A study of the effects of two factors,A and B,on an outcome Y was conducted.Factor A had three levels and Factor B had two.All six combinations of Factors A and B had the same number of observations.A two-way ANOVA model with interaction effects was employed.Part of the ANOVA table is given below.
(a) Write down the two-way factor effects model for the study.Specify the model assumptions. [3marks]
(b) Fill in the blanks marked by"?"in the ANOVA table. [6 marks]
(c) Test at the 5%level of significance for the interaction effects between the two factors. [3marks]
(d) Test at the 5%level of significance for the main effect of Factor A. [3marks]
(e) The marginal means of Y for the three levels of Factor A are given in the following. Construct a 95%confidence interval for
e = μ1. 一 μ2.+μ3.,
where μi.is the mean for Y for level i=1,2,3 of Factor A.
[8 marks]
[Total:23 marks]
5.
(a) Consider a linear regression model
Y=Xβ+E,
where Xis of dimension n×p,Y of dimension n×1 and e is a vector of n variables which have means 0 but are not necessarily independent among each other.Write down the least square estimate,β,for β in terms of Xand Y.No proof is required. [1 mark]
(b) Xis partitioned as
X=[X₁|X₂],
β as
and the least squares estimate β as
where X₁is the jth column (not necessarily the first column)of X,X₂is the matrix of Xwithout the jth column,β₁is the jth regression coefficient and β₂ is the vector of the remaining regression coefficients.Let ey be the residual vector obtained by regressing Yon X₂and e₁be the residual vector obtained by regressing X₁on X₂.Consider a model
ey=γe₁+ξ
Prove that the least squares estimate for γ is β1.
[10 marks]
[Total:11 marks]