STA 6207 – Exam 3 – Fall 2012 PRINT Name _____________________ All Questions are based on the following 2 regression models, where SIMPLE REGRESSION refers to the case where p=1, and X is of full column rank (no linear dependencies among predictor variables). Model 1 : Yi 0 1 X i1 p X ip i i 1,..., n i ~ NID 0, 2 Model 2 : Y Xβ ε X n p' β p'1 ε ~ N 0, 2 I _______________________________________________________________________________________ CONDUCT ALL TESTS AT = 0.05 SIGNIFICANCE LEVEL. Q.1. The mean and variance of the Poisson distribution are equal. That is: . Use Bartlett’s method to obtain a transformation to make the variance approximately constant. Q.2. Random Coefficient Regression: Yij 0i 1i X ij ij Derive V{Yij}. 0i 02 01 0i ~ NID , 2 1i 1i 01 1 ij ~ NID 0, 2 0i ij 1i Q.3. A regression through the origin is to be fit, relating Y to X for n=5 observations. The X levels are (1,2,3,4,10). p.3.a. Obtain X’X, (X’X)-1, the P matrix, and the diagonal elements of P (vii). p.3.b. What do the vii elements sum to? Which, if any, elements are potentially influential? n ^ p.3.c. Below is the fit for all cases (top), and with observation 5 dropped (bottom) b1 1 XY i 1 n X i 1 X Y 1 2 3 4 10 Y-hat 2 5 6 7 30 Observation 5 dropped X Y Y-hat 1 2 2 5 3 6 4 7 ^ e 2.75 5.51 8.26 11.02 27.54 2 i b1 -0.75 -0.51 -2.26 -4.02 2.46 e 1.93 3.87 5.80 7.73 i i 2.75 SS(Res) SS(Res) = s= b1 0.07 1.13 0.20 -0.73 All Cases: 1.93 SS(Res) Observation 5 dropped: SS(Res(5)) = s(5) p.3.d. Compute Y 5(5) , and using s(5)as estimate of : DFFITS5 , and DFBETAS1(5) n Note: ei2 0 i 1 ^ ^ Q.4..For model 1, a simple linear regression model (p=1), derive Cov 0 , 1 completing the following parts: ^ p.4.a. Write 1 n ^ n aiYi and 0 bYi i stating explicitly what the ai and bi values (functions) are i 1 i 1 ^ ^ p.4.b. Using rules of Covariances of linear functions of random variables to derive Cov 0 , 1 p.4.c. Researchers in the U.S. fit regressions of relationship between weight (Y) in pounds, and height (X) in inches, while foreign researchers work with weight (Y) in kilograms, and height (X) in centimeters. Give the Conversion Rates are as follow: 1 kilogram = 2 .2 pounds 1 cm = 2.54 inches Suppose that in a population, the standard deviation of weights is 22.0 pounds, and in a sample, the heights are given below: Run# X(Inches) X(Cms) 1 100 254 ^ 2 200 508 3 300 762 4 400 1016 ^ Cov 0 , 1 U.S. = ________________________ Foreign = __________________________ Q.5. A regression model is fit, relating January mean temperature (Y) to ELEVation, LATitude, and LONGitude for n=369 weather stations in Texas (the data are aggregated over a period of years). The following table gives the Regression and Residual sums of squares for each model. All models contain an intercept. Model ELEV LAT LONG ELEV,LAT ELEV,LONG LAT,LONG ELEV,LAT,LONG SS(REG) SS(RES) 5785.3 7986.3 12603.3 1168.3 2239.5 11532.1 13155.1 616.5 7669.7 6101.9 13087.0 684.6 13156.8 614.8 p.5.a. Compute R(LONG), R(LONG|ELEV), R(LONG|LAT), R(LONG|ELEV,LAT) p.5.b. Based on the simple linear regression, relating Y to LONG: E{Y} = 0 + LONGLONG, test H0: LONG = 0 HA: LONG ≠ 0 Test Statistic: ____________________________ Rejection Region: ________________________ p.5.c. Based on the simple linear regression, relating Y to ELEV, LAT, LONG: E{Y} = 0 + ELEVELEV+ LATLAT + LONGLONG, test H0: LONG = 0 HA: LONG ≠ 0 Test Statistic: ____________________________ Rejection Region: ________________________ p.5.d. The Coefficient of Determination, when LONG is regressed on ELEV and LAT is 0.843. Compute the Variance Inflation Factor for LONG.