Open book and notes. 120 minutes. Cover page and six pages of exam. No calculators.

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1 IE 330 Seat # Ope book ad otes 120 miutes Cover page ad six pages of exam No calculators Score Fial Exam (example) Schmeiser

2 Ope book ad otes No calculator 120 miutes 1 True or false (for each, 2 poits if correct, 1 poit if left blak) (a) T F I hypothesis testig, we assume that H 0 is true ad ask whether the data are cosistet with that assumptio (b) T F I regressio aalysis, we explai variability i the depedet variable by relatig it to idepedet variable(s) (c) T F I cofidece itervals, the cofidece arises from the procedure, ot from the particular iterval (d) T F The F distributio is used to compare variaces (e) T F The T distributio is used to compare meas (f) T F If X ad Y have zero correlatio, the Cov(X, Y ) is ot defied (g) T F The acroym "iid" meas "idepedet ad idetically distributed" (h) T F Decreasig the type-oe error probability icreases the power of the test 2 (Motgomery ad Ruger, Problem 11 1) A article i Cocrete Research ("Near Surface Characteristics of Cocrete: Itrisic Permeability," 41, 1989), preseted data o comprehesive stregth x ad itrisic permeability y of various cocrete mixes ad cures Summary quatities are = 14, Σ y i = 572, Σ y i2 = 23,530, Σ x i = 43, Σ x i2 = ad Σ x i y i = Cosider the regressio model y i =β 0 +β 1 x i +ε i (For Parts (a) ad (b), you do ot eed to solve for a umerical aswer, but ayoe with a calculator should be able to fiish your calculatio) (a) Write the equatio for the least-squares estimated value of the y itercept Σ y i Σ x i βˆ 0 = y βˆ 1 x, where y =, x =, adβˆ 1 is defied i Part (b) (b) Write the equatio for the estimated value ofβ 1 (Σ y i )( Σ x i ) βˆ 1 = Σ x i y i / (Σ x i )2 Σx i2 Fial Exam (example) Page 1 of 6 Schmeiser

3 3 Suppose that y i =β 0 +β 1 x i +β 2 x i 2 +εi Letβ j be estimated byβˆ j for j = 0, 1, 2 (a) For a future observatio, say x = 3, write the predicted value of y? ŷ = βˆ 0 + 3βˆ 1 + 9βˆ 2 Commet: The predicted value used estimated coefficiets ad has o error term (b) For a observatio x i, write the correspodig residual y i ŷ = y i (βˆ 0 +βˆ 1 x i +βˆ 2 x i2 ) 4 (Motgomery ad Ruger, third editio, page 404) Cosider the results from the blue box, which icludes iformatio about the regressio-coefficiet estimators ad the associated ANOVA (a) What is the value of SS R? Directly from the Table, read (b) Is SS R a radom variable? (That is, if the experimet were repeated by replacig the curret sample with aother, would SS R chage value?) yes o (c) Is the residual-error degrees of freedom a radom variable? yes o (d) Are the three P values radom variables? yes o (e) The value ofβˆ 1 is 290 Do we kow the value ofβ 1? yes o (f) Does this aalysis idicate that legth ad stregth are idepedet of each other? yes o Fial Exam (example) Page 2 of 6 Schmeiser

4 5 Cosider the equatio for the least-squares estimates βˆ = (X X) 1 X y from page 417 of Motgomery ad Ruger, third editio (a) What is the otatio for the vector of resposes? y (b) Explai why the first colum of X is all oes The costat "1" is the coefficiet ofβ 0 i the liear model The first colum of X correspods toβ 0 Each row of X correspods to oe of the observatios Therefore, the colum of oes is the coefficiets ofβ 0 (c) Explai the meaig of E(βˆ)=β Each of the poit estimators of theβcoefficiets is ubiased Commet Various correct aswers could be give (d) The variace σ 2 is estimated usig σˆ 2 / ( p ) What radom variable is σ 2 the variace of? Each of the additive radom-error terms,ε i, for i = 1, 2,, Fial Exam (example) Page 3 of 6 Schmeiser

5 6 Cosider the desig of experimets statistical model Y ij =µ+τ i +ε ij for i = 1, 2,, a ad j = 1, 2,, (a) Explai the meaig ofτ 1 The first treatmet effect More specifically, i the model the respose is a liear combiatio, so E(Y 1j )=E(µ+τ 1 +ε ij )=µ+τ Therefore, τ 1 = E(Y 1j ) µ, where E(Y 1j ) is the expected respose coditioal that treatmet oe is received adµis the grad mea of all resposes (b) Explai the meaig of a The umber of treatmets is a (This is also the umber of levels of the sigle factor) (c) How may factors are i this experimet? oe Fial Exam (example) Page 4 of 6 Schmeiser

6 7 Cosider the statistic (X µ 0 ) / (S / ) (a) What is the stadard error of X? ste(x ) = V(X i ) / Commet: The estimated stadard error is ste ˆ (X ) = S 2 / (b) If X ad S 2 are computed from a iid sample ad the ull hypothesis H 0 :µ=µ 0 is true, the what is the distributio of the statistic? Studet T, with 1 degrees of freedom Commet: Studet T is a family of distributios Specifyig the degrees of freedom yields a particular distributio (c) If the sample size grows, the E(S 2 ) decreases remais uchaged icreases (d) This statistic is useful for cofidece itervals hypothesis testig both Fial Exam (example) Page 5 of 6 Schmeiser

7 8 Motgomery ad Ruger (Problem 10 5) Two machies are used to fill plastic bottles with dish-washig deterget The stadard deviatios of fill volumes are kow to be σ 1 = 010 fluid ouces ad σ 2 = 015 fluid ouces for the two machies, respectively Two radom samples of 1 = 12 bottles from Machie 1 ad 2 = 10 bottles from Machie 2 are selected, ad the sample mea fill volumes are x 1 = 3087 ad x 2 = 3068 (a) Determie the stadard error of X 1 X 2 V(X 1 X 2 ) = 1 2 V(X 1 )+( 1) 2 V(X 2 ) idepedece = V(X 1 )+V(X 2 ) simplify σ = 1 σ 2 2 variaces of sample meas = give values Takig the square root yields the stadard error (b) The textbook s problem begis by askig for a 90% two-sided cofidece iterval o the mea differece i fill volume The questio here is to defie "mea differece i fill volume" Let µ 1 deote the mea fill volume for machie 1 Let µ 2 deote the mea fill volume for machie 2 The the mea differece i fill volume is the costat µ 1 µ 2 (c) The 90% cofidece iterval from Part (b) is (00987, 02813) Suppose that this were a test of hypothesis cosiderig H 0 :µ 1 =µ 2 versus H 1 :µ 1 =/ µ 2 If the probability of type-oe error isα=010, would we reject H 0? yes o ot eough iformatio to kow Fial Exam (example) Page 6 of 6 Schmeiser

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet