hw5.sol - Page 1 of 8 Stat209/Ed260 D Rogosa 2/20/09...

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Stat209/Ed260 D Rogosa 2/20/09 Solutions Assignment 5. The many uses and forms of analysis of covariance Problem 1 The prediction equation to work from is S = 1900 + 230*B + 18*A + 100*E + 490*D + 190*Y + 50*T - 2400*X The coefficient of T in the regression equation is 50. Note that salary is in an arbitrary 'funny-money' unit. It tells us that there will be a 50 unit decrease in salary for someone whose student evaluations change from very good (T=1) to poor (T=0). This is true no matter what the values of the other variables are (of course this conclusion applies only to this model, with its set of predictors). partial regression coefficient of T here could be interpreted as the average change in salary between the two levels of student evaluation, which is 50. -------------------- To do this problem the HARD way, you can plug in the values of the other variables that the question provides for us (B=A=E=D=X=1 and Y=5) into the regression equation: S = 1900 + 230(1) + 18(1) + 100(1) + 490(1) + 190(5) + 50(T) - 2400(1) = 1288 + 50(T) For a professor with good student evaluations (T = 1): S = 1288 + 50(1) = 1338 For a professor with bad student evaluations (T = 0): S = 1288 So the expected change in salary is 1288-1338 = -50. (That is, a decrease of 50). ------------------------------------ Part b. model for the S on X regression is E(S|X) = beta(0) + beta(1)*X To calculate the value of the estimate of the slope (beta(hat)(1)), first look at the expected values of X within each group and realize that they are equal to the means of S within each group. sample value of E(S|X=0) = beta(hat)(0) = mu(hat)(0) E(S|X=1) = beta(hat)(0) + beta(hat)(1) = mu(hat)(1) Where: mu(hat)(0) = the mean salary for males = 16,100 mu(hat)(1) = the mean salary for females = 11,200 To calculate the slope, beta(hat)(1), we subtract the two group means: beta(hat)(1) = mu(hat)(1) - mu(hat)(0) = 11,200 - 16,100 = -4900 Another way to do this problem is to realize that the line for the regression of S on X will pass through the points (0, 16100) and (1, 11200). The slope = (change in S)/(change in X) = (16100 -11200)/(0 - 1) = -4900. ----------------------------------------------------------------- Problem 2 Page 1 of 8 06/12/2010 http://www-stat.stanford.edu/~rag/stat209/09hw5.sol
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carry out standard ancova > incent = read.table(file="D:\\drr06\\stat209\\retention.dat", header = T) > attach(incent) > summary(incent) Retent StudyMin Incent Min. : 3.000 Min. : 5.00 Min. :0.0 1st Qu.: 5.750 1st Qu.: 8.75 1st Qu.:0.0 Median : 8.000 Median :12.50 Median :0.5 Mean : 7.875 Mean :12.50 Mean :0.5 3rd Qu.:10.000 3rd Qu.:16.25 3rd Qu.:1.0 Max. :13.000 Max. :20.00 Max. :1.0 > anc = lm(Retent ~ Incent + StudyMin) > summary(anc) Call: lm(formula = Retent ~ Incent + StudyMin) Residuals: Min 1Q Median 3Q Max -2.5083 -0.7833 -0.1000 0.8792 1.6750 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.87500 0.65165 4.412 0.000243 *** Incent 4.08333 0.49260 8.289 4.64e-08 *** StudyMin
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This note was uploaded on 08/20/2011 for the course STATS 209 taught by Professor Rogosa,d during the Winter '09 term at Stanford.

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hw5.sol - Page 1 of 8 Stat209/Ed260 D Rogosa 2/20/09...

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