9. Simple linear regression G2.1) Show that the vector of residuals e = Y Ŷ has the covariance matrix (I X(X T X) 1 X T )σ 2.

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1 LINKÖPINGS UNIVERSITET Matematiska Istitutioe Matematisk Statistik HT TAMS24 9. Simple liear regressio G2.1) Show that the vector of residuals e = Y Ŷ has the covariace matrix (I X(X T X) 1 X T )σ 2. G2.2) Cosider the simple liear regressio model Y j = β 0 + β 1 x j + ε j, j = 1, 2,..., where ε 1, ε 2,..., ε are idepedet N(0, σ 2 ). Show that the least square estimates ˆβ 0 ad ˆβ 1 are idepedet if ad oly if x j = 0. j=1 G2.3) A ew medicie agaist cacer was tested o te mice, each with a tumor of the size 4 grams. The mice were give dieret doses (x) of the medicie ad the reductio (y) of the tumor was measured o each mouse. Result: x y Model: Y = β 0 + β 1 x + β 2 x 2 + ε where ε = N(0, σ 2 ). A aalysis of variace accordig to this model gave: Aalysis of variace Estimated regressio lie: y = x x 2 i ˆβi d( ˆβ i ) REGR RES TOT /5

2 a) Test H 0 : β 1 = β 2 = 0 agaist H 1 : at least oe of β 1 ad β 2 0, o the level. b) Fid the optimal dose accordig to this regressio aalysis. c) Is it reasoable to remove oe of the explaatory variables? Briey justify your aswer. G2.4) I a study for the protability of movie studies, 20 Hollywood movies were chose radomly ad for each movie the followig values were obtaied: y = gross reveue (uit: millios of USD) x 1 = productio cost (uit: millios of USD) x 2 = marketig cost (uit: millios of USD). There was special iterest i cosiderig whether there was ay iuece if the movie was based o a book that had bee published before the movie was produced. To separate such movies from the others a so called dummy-variable was deed Result: x 1 x 2 x 3 y x 3 = { 1 for movies based o a book 0 otherwise Data has bee aalyzed accordig to the models Model 1: Y = β 0 + β 2x 2 + ε Model 2: Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + ε where ε respectively ε i the two models are idepedet ad N(0, σ 2 ) respectively N(0, σ 2 ). Aalysis of variace ca be foud below. Aalysis of variace o. 1 Estimated regressio lie: y = x 2 2/5

3 i ˆβi d( ˆβ i ) REGR RES TOT Aalysis of variace o. 2 Estimated regressio lie: y = x x x 3 i ˆβi d( ˆβ i ) REGR RES TOT a) Briey explai why model 2 explais the data better tha model 1. Justify your aswer with suitable parameters from the data aalysis. b) Does seem to aect the reveue if the movie is based o a book ad i that case i which way is it aected? Justify your aswer with a suitable 95% codece iterval. c) A movie based o a just published book is about to be produced. The productio cost is estimated to be 11 millio USD ad 9 millio will be spet o marketig the movie. Estimate the expected gross reveue from the movie usig model 2. You do't have to costruct a iterval (X T X) 1 = G2.5) Suppose that the time Y for a chemical reactio has a liear regressio with respect to the temperature x. We have the followig data: x i y i a) Calculate poit estimates for β 0, β 1 ad σ 2. b) Plot the data poits as well as the regressio lie i a coordiate system. 14.2) For the umerical data (x 1, y 1 ), (x 2, y 2 ),..., (x 10, y 10 ) the followig has bee calculated xi = 12.0, x 2 i = 18.40, yi = 15.0, y 2 i = 27.86, xi y i = The data is described by a regressio model y i = α + βx i + z i where z 1,..., z 10 are idepedet observatios from N(0, σ). Fid 95% codece itervals for β ad α. 3/5

4 14.4) To see if a certai dimesio y at a maufactured item depeds o the settig x o a certai machie, y was measured for 7 dieret settigs of the machie ad the followig data was obtaied: x : y : a) Plot the data i a coordiate system ad determie if it is reasoably to assume that y depeds liearly o x (with some radom variatios). b) Estimate the parameters of the regressio model ad plot the regressio lie i the coordiate system. c) If you wish to have 2.5 as the expected value for the dimesio, what should the settig of x be? d) Fid a 95% codece iterval for the itercept α ad the slope β. e) Fid a 95% codece iterval for µ 0 = α + βx 0 ad plot the boudaries as well as the estimated regressio lie i a coordiate system. 14.7) A testig facility had the task to ivestigate how the icotie cotet (y) depeds o the cotets of carbo (x 1 ) ad chloride (x 2 ). A multiple regressio model approach was made y = α + β 1 x 1 + β 2 x 2 + ε where ε is N(0, σ). After computer processig the data the followig table was obtaied: Coeciet Stadard error p-value Lower 95% Upper 95% Itercept Carbo E Chloride I additio the covariace betwee the β 1 - ad the β 2 - estimates was estimated to a) Should the hypothesis H β1 : β 1 = 0 respectively H β2 : β 2 = 0 be rejected o the 5% sigicace level? b) Estimate how much the expected icotie cotets is chaged if both the carbo- ad chloridecotet is icreased by 1 uit. c) Calculate the stadard error for the estimate i b). 4/5

5 ANSWERS G2.1 Write e as a liear trasformatio of the Y vector. G2.3 a) v = > (from F (2, 7)-table). H 0 is rejected. b) x = 5.82 gives a maximum accordig to the estimated regressio equatio. c) Takig ito accout the appearace of the curve, oe of the explaatory variables should be excluded. G2.4 a) Model 2 has the coeciet of determiatio R 2 = 96.7% which is sigicatly better tha R 2 = 77.9% for model 1. b) I β3 = ( ) = (3.312, ). It seems that a mauscript based o a book gives higher gross reveue. c) ˆβ ˆβ ˆβ 2 + ˆβ 3 = The expected gross reveue is approximately 67 millio dollars. G2.5 a) Y = Xβ + ε where ( ) X T =, y T = ( ), ( ) 1 ( ) ˆβ = (X T X) 1 X T y = = ( ) ( ) 47.5 = ˆβ 0 + ˆβ 1 x i : σ 2 is estimated by s 2 = 1 3 (yi ˆβ 0 ˆβ 1 x i ) 2 = I β = respectively I α = ( a) It seems reasoable to assume that y depeds liearly o x; b) α = , β = ; c) x = 3.66; d) I α = α t p/2 ( 2) s 1 + x 2 i=1 (x gives I i x) 2 α = , I β = β t p/2 ( 2) s/ i=1 (x i x) 2 gives I β = ; e) I µ0 = µ 0 t 1 p/2( 2) s + (x 0 x) 2 i=1 (x. i x) a) The rst hypothesis is rejected but the other is ot sice the p-value is less tha 5% i the rst case but ot i the other; b) The expected icrease is β1 + β 2 = 1.247; c) V [β1 + β 2 ] = V [β 1 ] + V [β 2 ] + 2C[β 1, β 2 ] which is estimated with = The stadard error is the square root of this value d(β1 + β 2 ) = ) 5/5