SA3101 Final Examination Solution SA3101 Regression Analysis Solutions of the Final Examination Semester II, 2001-2002 (d), (c ), (b), (a), (d). (a) The matrix plot looks like (b) The matrix plot looks like 4/28/2019 SA3101 Final Examination Solution

SA3101 Final Examination Solution 2. (continued). (c ) The potential-residual plot is (d) n=25, p=4. H0: beta3=2 v.s. Beta3 not =2 , T3=-4.37, df=n-p-1=50-4-1=45. T(45,.025)=2.01. The graph indicating the rejection regions and T3 looks like Since T3 falls into the left sided rejection region, H0 is rejected at 5% level of significance. 4/28/2019 SA3101 Final Examination Solution

SA3101 Final Examination Solution 2. (continued) (d) H0: beta1=0, beta4=0 v.s. H1: H0 is false. F=2.1, df=(2,45). The graph indicating the rejection region and F is as follows: Since F doesn’t fall into the rejection region, H0 is not rejected. Thus, the reduced model is adequate. (e). The index plot of the Cook’s distance is 4/28/2019 SA3101 Final Examination Solution

SA3101 Final Examination Solution 3. (a). We should use the studentized residual here. The following is the index plot of the studentized residuals: Observations 4,5, 7, 19 are outliers, using cut-off value 2. (b). Observations 4 and 5 are high leverage points, using cut-off value 2(p+1)/n=.5 for Pii. (c ). Observations 4,5, 7, 19 are influential points, using cut-off value 2((p+1)/(n-p-1))^.5=1.1546. (d). The rough potential-residual plot is 4/28/2019 SA3101 Final Examination Solution

SA3101 Final Examination Solution 4. n=48, p=5, R^2(F)=30.7% (a). R0^2=0. R^2 reflects the proportion of the total variability explained by the predictors. Since there are no predictors in the model, the R^2=0. (b). X3(Taxes) will be first introduced since R^2=16.6% is the largest R^2 among all R^2 for single predictor models. H0: Y=beta0+e v.s. H1: Y=beta0+X3beta3+e. F=(R3^2-R0^2)/1/(1-R3^2)*(48-2)=.166/.834*46=9.1559, df=(1,46), F(.05,1,46)=4.04. X3 should be introduced into the model. Thus, the variable “Taxes” is the first factor that affects the Domestic immigration behavior of American people. (c) . X1 (crime), X3 (Taxes) and X4 (Educ.) should be first introduced since the associated R^2 is the largest one among all R^2 with 3-variable models. H0: Y=beta0+X3beta3+e v.s. H1: Y=beta0+X1beta1+X3beta3+X4beta4+e R^2(F)=.307, df(F)=48-3-1=44; R^2(R )=.166, df(R )=48-1-1=46 F=(.307-.166)/2/(1-.307)*44=4.476, df=(2,44), F(2,44,.05)=3.19, Reject H0, the reduced model is not adequate. That is, X1 and X4 should be introduced. Thus, Crime, Taxes and Education are three major concerns when American people consider domestic immigration. (d). Either X5 or X2 should be first deleted. This is because R^2 reduces almost 0 if X5 or X2 is deleted. 4/28/2019 SA3101 Final Examination Solution

SA3101 Final Examination Solution 4. (continued) (e) Model “34” is the best model since the associated R^2 is the largest among all R^2 with 2-variables. Note that X’s 13 23 24 34 35 R^2 16.8% 19.5% 8.5% 26.7% 17.4% Ra^2 13.5% 15.9% 4.43% 23.4% 13.7% Thus, according to adjusted R^2, model “34” is also the best. Actually Ra^2=1-(n-1)/(n-p-1)*(1-R^2) This means that for the same p, Ra^2 is strictly increasing with R^2. This is the main reason why the solution is the same. (f). p 0 1 2 3 4 5 R^2 0 16.6% 26.7% 30.7% 30.7% 30.7% Ra^2 0 14.8% 23.4% 26.0% 24.3% 22.45% Since for p=3,4,5, the R^2 changes almost 0 while from p=2 to p=3, the change is big. Thus, p=3 is the best model. The associated adjusted R^2 confirm this conclusion. 4/28/2019 SA3101 Final Examination Solution