[4+3+3] Q 1. (a) Describe the normal regression model through origin. Show that the least square estimator of the regression parameter is given by

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1 Concordia University Department of Mathematics and Statistics Course Number Section Statistics 360/1 40 Examination Date Time Pages Final June hours 7 Instructors Course Examiner Marks Y.P. Chaubey Y.P. Chaubey 60 Special Instructions: CLOSED BOOK EXAM 1. Calculators are permitted. 2. Full Credit will be given only for systematic and detailed work. 3. Tables needed are given on the last page 4. Answer ALL questions. [4+3+3] Q 1. (a) Describe the normal regression model through origin. Show that the least square estimator of the regression parameter is given by b = ni=1 X i Y i ni=1. Xi 2 (b) Show that where b N(β, σ 2 {b}) σ 2 {b} = σ 2 ni=1 X 2 i (c) Prove the Bonferroni inequality P (A C 1 A C 2 ) 1 P (A 1 ) P (A 2 ) and hence justify that the region in the (β 0, β 1 ) plane given by β i b i Bs{b i }, i = 0, 1, where B = t(1 α 4 ; n 2) provides a 100(1 α)% joint confidence region for (β 0, β 1 ) in context of the simple linear model.

2 Stat 360/1 Final Examination June 2004 Page 2 of 6 [4+4+2 ]Q 2. A substance used in biological and medical research is shipped by airfreight users in cartons of 1,000 ampules. The data below, involving 10 shipments were collected on the number of times the carton was transferred from one aircraft to another over the shipment route (X) and the number of ampules found to be broken upon arrival (Y ). i : X i : Y i : Assume that a first order regression model is appropriate to answer the following questions. (a) Set up the matrices X and Y and compute the estimator of the regression parameter using the matrix methods; b = (X X) 1 X Y = [ ] (b) Use matrix methods to reproduce the following ANOVA table for the above data: Analysis of Variance Source DF SS MS F Regression Residual Error Total (c) Determine s 2 {b 0 },s 2 {b 1 } and s{b 0, b 1 } using the matrix formula for s 2 {b}. Use these to compute s 2 {Ŷh} for X h = 0 and show that it coincides with MSE(1 + X h(x X) 1 X h ) where The matrix X X should equal X h = X X = [ 1 0 ]. [ ]

3 Stat 360/1 Final Examination June 2004 Page 3 of 6 [4+4+2 ]Q 3. The following are the sample data provided by a moving company on the weights of six shipments, the distances they were moved and the damage that was occured. Weight(1,000 lbs) Distance(1,000 miles) Damage($) i X i1 X i2 Y i Assume that the regression model Y i = β 0 + β 1 X i1 + β 2 X i2 + ɛ i fits the data. Using the method of matrices, obtain the following: (i) Vector of estimated Regression Coefficients, i.e. b. (ii) SST O, SSR and SSE. (iii) Estimated Variance Covariance Matrix of b You can use the values of (X X) 1 and (X Y ). as given below: (X X) 1 = , (X Y ) = [5+5 ]Q4. (a) Let Y be a random vector with n components and A be a matrix of constants of order m n, using the matrix methods, show that the mean and variancecovariance matrix of W = AY is given by E{W} = AE{Y} σ 2 {W} = Aσ 2 {Y}A (b) Use the above results to prove the following; (i) E{e} = 0. (ii) σ 2 {e} = σ 2 I H, where H = X(X X) 1 X. 3

4 Stat 360/1 Final Examination June 2004 Page 4 of 6 [3+4+3 ]Q 5. In a marketing research study the relation between brand liking (Y ) and moisture content (X 1 ) and sweetness (X 2 ) of the product, the following results were obtained, i X i X i Y i (a) The MINITAB output after running a regression of Y on X 1 and X 2 is given below. The regression equation is Y = X X2 Predictor Coef StDev T P Constant X X S = R-Sq = 91.9% R-Sq(adj) = 90.1% Analysis of Variance Source DF SS MS F Regression Residual Error Total Source DF Seq SS X X

5 Stat 360/1 Final Examination June 2004 Page 5 of 6 Answer the following questions based on the above output. (i) Use extra SS principle to test if X 2 should be retained in the model at α = (ii) Perform a lack of fit test for the appropriateness of the two predictor first order linear model (use α =.05. (b) After running the two variable regression above, the researcher thought that it might be better to incorporate the interaction term in the model and thus fitted a regression of Y on X 1, X 2 and X 1 X 2 and produce the following ANOVA table; [5+5 ]Q 6. Analysis of Variance Source DF SS MS F Regression Residual Error Total Use Extra SS principle to determine if the term X 1 X 2 should also be included given that X 1 and X 2 are already in the model. Use a level of significance α = Use the the data and output of the Q 5 (a) to answer the following questions assuming that the two predictor first order linear model is appropriate. Use (X X) 1 = (a.) Find joint confidence intervals for E{Y h } for X h1 = 8, X h2 = 2 and for X h1 = 8, X h2 = 4 with joint confidence coefficient=.90, using Bonferroni and Working -Hotelling approaches. Which of these will be preferred. (b) If two new observations for Y obtained at the above two prescribed levels of X 1 and X 2 are 80 and 90. Would you have 90% confidence in claiming that these observations are generated from the first order model estimated in Q 5 (a)? 5

6 Stat 360/1 Final Examination June 2004 Page 6 of 6 Values of t(a; ν) A ν Values of F (A; ν 1, ν 2 ) A ν 1 = 2, ν 2 = 6 ν 1 = 3, ν 2 = 6 ν 1 = 2, ν 2 = 9 ν 1 = 3, ν 2 = 9 ν 1 = 2, ν 2 =