Baruch College, STA 9700 Sample Final Page |1 Baruch College, STA 9700 Sample Final Page |2 Coefficients : 1. For each of the following regression models, indicate whether it is a general linear regression model. Estimate Std. Error t value Pr (>|t/) If it is not, state whether it can be expressed in the form of (Intercept) 158. 4913 18 . 1259 X1 -1. 1416 0. 2148 Yi = Bo + BIXi1 + B2Xi2 + ...+ Bp-1Xi,p-1+ Ei X2 -0. 4420 0. 4920 X3 -13. 4702 7. 0997 via a suitable transformation. --- Signif. codes: 0 '***' 0.001 \\* *' 0.01 * *' 0.05 . .' 0.1 " ' 1 (A) (5 points) Yi = Bo + BiXi + BlogXiz + B3Xi + &; Studentized residual (B) (5 points) Yi = ; exp(Bo + BiXi + B2X12) 2. In a study of mercury level in 150 Maine lakes, a regression model was developed with 2 predictors: the type of lake and the elevation of lake. The three lake types considered were A = oligotrophic, B = eutrophic, and C = mesotrophic. Studentized residual Sample Quantile (This predictor is used for lake classification and is related to the types of organisms found in the water.) The type of lake was coded as follows Type of lake X2 X3 A = oligotrophic 40 50 60 70 90 B = eutrophic Predicted Theoretical Quantiles C = mesotrophic Analysis of Variance Table (A) (3 points) Write the first-order linear regression model relating mercury level (Y) of a lake to its Response : Y elevation (X1) and type (X2, X3). Df Sum Sq Mean Sq F value Pr (>F) X1 1 8275.4 8275.4 (B) (6 points) Write the sub-models for the three lake types, respectively. X2 1 480.9 480.9 X3 364.2 364.2 (C) (6 points) Interpret B2 and B2 - 3, respectively, in the context of the problem. Residuals 42 4248.8 101.2 --- 3. Suppose Y1, Yz, and Y3 are independent standard normal random variables. Define the random vector Signif. codes: 0 '* **' 0 0.001 ' * * 0.01 * 0.05 .' 0.1 '' 1 (W1) (W3/ W = (W2 , where W1 = Y1 + Yz + Y3, W2 = Y1 - 212, and W3 = 2Y1 - Y2 - Y3- Anova Table (Type III tests) Response : Y Sum Sq Df F value Pr (>F) (Intercept) 7734.5 (A) (3 points) Find the expectation and the variance-covariance matrix of Y = X1 2857.6 X2 81.7 1 364.2 1 B) (3 points) Find the matrix A in the expression W = AY. Residuals 4248.8 42 (C) (2 points) Find the expectation of W. Signif. codes: 0 '***' 0.001 \\ **' 0.01 '* 0.05 . .' 0.1 . ' 1 D) (4 points) Find the variance-covariance matrix of W. (Hint: you may use the formula o {W} = Variables in Ao {Y}A'.) R2 AICp PRESSp model None 262.916 88.16 13,970.10 4. A hospital administrator wished to study the relation between patient satisfaction (Y) and patient's age (X1, in years), severity of illness (X2, an index), and anxiety level (X3, an index). Data were X1 6190 220.529 8.35 5,569.56 3635 244.131 42.11 ,254.49 collected on randomly selected 46 patients, where larger values of Y, X2, and X3 are, respectively, X 2 associated with more satisfaction, increased severity of illness, and more anxiety. A multiple X3 4155 240.214 35.25 8,451.43 regression model containing all 3 predictor variables as first-order terms was fitted to the data. The X1, X. 6550 217.968 5.60 5,235.19 partial R outputs are given below. (Note: use Table B.2 on pages 1317-8 and Table B.4 on pages X1 , X 6761 215.061 2.81 4,902.75 1320-6 of the textbook to find relevant critical values.) X2, X3 4685 237.845 30.25 8,115.91 X1,X2, X3 6822 216.185 4.00 5,057.89