Statistics 333 Assignment 1 Due Sept. 19, 20031. For the simple linear regression model Yi=β0+β1Xi+εi, i = 1, . . . , n,consider the leastsquares fitsˆYi= b0+ b1Xiand residuals givenbyei=Yi−ˆYi=Yi−(b0+b1Xi),i=1, . . . , n.Since b0= Y − b1X,itiseasy to see (1/n)Σni=1ˆYi= Y and hence also thatΣni=1(Yi−ˆYi) = 0.Similarly,also showthat the residuals eisatisfy the additional ‘orthogonality’ constraintΣni=1Xi(Yi−ˆYi) = 0.2. For the simple linear regression model in Exercise 1, under the standard model assump-tions for the random errorsεi,showthat the least squares estimator b1= Sxy/SxxandY = (1/n)Σni=1Yihave zero covariance, i.e., Cov( b1, Y ) = 0 ,where Sxy=Σni=1(Xi− X) Yi.3. For the simple linear regression model, the estimator of the error varianceσ2= Va r(εi) isgivenbyS2=1n−2ni=1Σ(Yi−ˆYi)2≡1n −2SSE .Showthat this estimator S2is unbiased forσ2,i.e., prove that E( S2) =σ2.Note: Toprove this result, use the following approach. First, verify the identityYi− (β0+β1Xi) = [ Yi− (b0+ b1Xi) ] + [ Y − (β0+β1X) ] + [ (b1−β1)(Xi− X) ] .Then verify that the sum of squares of elements on the left-hand side,Σni=1[ Yi− (β0+β1Xi) ]2,isequal to the sum of the 3 sums of squares of individual elementson the right-hand side, due to ‘orthogonality’ in cross-terms (Exer.1). Hence, verify thatΣni=1[ Yi− (β0+β1Xi) ]2= SSE + n[ Y − (β0+β1X) ]2+ Sxx(b1−β1)2.Finally,equate the expected values of both sides of the sums of squares relation, to getnσ2= E[SSE ] + nE[ Y − (β0+β1X) ]2+ SxxE[ (b1−β1)2], ‘evaluate’ the other expectedvalue terms, and solvefor E[SSE ].Also use definitions and known results for Va r( b1)etc., for instance, E{[ Yi− (β0+β1Xi) ]2} = Va r(Yi) ≡σ2by definition of variance.4. Consider the zero or no intercept model givenbyYi=β1Xi+εi,i=1, . . . , n,with the errorsεibeing independent, normal r.v.’ s with mean 0 and varianceσ2.i) Derive the least squares estimator b1ofβ1,and also derive the variance of this estimator.ii) For an arbitrary fixed value X0of X,establish that a 100(1 −α)%confidence interval forthe mean response value E( Y | X0) =β1X0is givenbyb1X0±t(α/2)n−1S√  X20/Σni=1X2i,where S2=Σni=1(Yi− b1Xi)2/(n − 1) provides an unbiased estimator ofσ2with n − 1 df.Note: You can conclude that ( b1−β1)/ se( b1) has the tn−1distribution.5. An experiment was conducted to study the mass of a tracer material exchanged betweenthe main flowofanopen channel and the "dead zone" caused by a sudden open channelexpansion. Researchers need this information to improve the water quality modeling capa-bility of a river. Itisimportant to determine the exchange constant K for varying flowconditions. The value of K describes the exchange process when a dead zone appears. Inastudy,values of the Froude Numbers (NF)were used to predict K .Numbers are func-tions of upstream channel velocity and water depth. The data collected were as follows,with the negative sign of the K values indicating "flushing", the direction of mass transferout of the dead zone:Obs NFK10.012500 -0.1256220.023750 -0.1206230.025625 -0.0962540.030000 -0.0806250.033125 -0.0793760.038125 -0.0731270.038125 -0.0725080.038125 -0.0718790.041250 -0.0937510 0.043125 -0.0531211 0.045000 -0.0575012 0.046875 -0.0575013 0.047500 -0.0500014 0.050000 -0.0437515 0.051250 -0.0700016 0.056250 -0.0512517 0.062500 -0.0312518 0.068750 -0.0387519 0.077500 -0.0168720 0.044000 -0.04625These data are stored in the file: /u/r/e/reinsel/stat333/exchange.dat ; also on webpage.Perform and showcalculations for (ii)-(v) belowby‘direct calculations’ on a calculator orcomputer; then use regression in Minitab or other software to confirm calculations.i) Construct a scatter plot of K versus NF,and provide some relevant comments.ii) Use the least squares method to fit the model Ki=β0+β1(NF)i+εi.iii) Compute S2, R2,and obtain the basic analysis of variance (ANOVA )table. Providesome brief interpretation for these results, e.g., in terms of the amount of variation of the Kvalues explained by the fitted regression (i.e., the variable NF).iv)Obtain the standard errors for the least squares estimates b0and b1,and under the usualnormal theory model assumptions, give a 95% confidence interval forβ1.v) Determine the explicit form for the 95% confidence interval of a mean responseE( K | NF) =β0+β1(NF) for anyfixedvalue NF,and evaluate for three selected and rea-sonable values of NF.vi) Obtain the residuals ei= Ki−ˆKi,and plot these versus the fitted valuesˆKias well asagainst the predictor variable (NF)i.Does this plot tend to confirm the basic assumptionsof the linear regression model, or does it suggest violation of anyassumption?