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Chapter 6

Chapter 6. Exercise 1. X=c(5,8,9,7,14) Y=c(3,1,6,7,19) R function ols ( x,y ) returns ( Intercept ) -8.477876 x ( slope ): 1.823009 mean(x )=8.6, mean(y )=7.2. Exercise 2. X=c(5,8,9,7,14) Y=c(3,1,6,7,19). Exercise 3. X=c(5,8,9,7,14) Y=c(3,1,6,7,19)

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Chapter 6

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  1. Chapter 6

  2. Exercise 1 X=c(5,8,9,7,14) Y=c(3,1,6,7,19) R function ols(x,y) returns (Intercept) -8.477876 x (slope): 1.823009 mean(x)=8.6, mean(y)=7.2

  3. Exercise 2 X=c(5,8,9,7,14) Y=c(3,1,6,7,19)

  4. Exercise 3 X=c(5,8,9,7,14) Y=c(3,1,6,7,19) The sum of squared residuals will be larger in this line relative to LSR because the LSR line is designed to minimize the residuals.

  5. Exercise 4

  6. Exercise 5 a=c(3,104,50,9,68,29,74,11,18,39,0,56,54,77,14,32,34,13,96,84,5,4,18,76,34,14,9,28,7,11,21,30,26,2,11,12,6,3,3,47,19,2,25,37,11,14,0) b=c(0,5,0,0,0,6,0,1,1,2,17,0,3,6,4,2,4,2,0,0,13,9,1,4,2,0,4,0,4,6,4,4,1,6,6,13,3,1,0,3,1,6,1,0,2,11,3) The R function ols(a,b)returns (Intercept) 4.58061839 x (slope) -0.04051423

  7. Exercise 6 c=c(300,280,305,340,348,357,380,397,453,456,510,535,275,270,335,342,354,394,383,450,446,513,520,520) d=c(32.75,28,30.75,29,27,31.20,27,27,23.50,21,21.5,22.8,30.75,27.25,31,26.50,23.50,22.70,25.80,27.80,21.50,22.50,20.60,21) Ols(c,d) yields: Higher levels of solar radiation predict lower rates of cancer.

  8. Exercise 7 a=c(500,530,590,660,610,700,570,640) b=c(2.3,3.1,2.6,3.0,2.4,3.3,2.6,3.5) R function ols(a,b) returns (Intercept) 0.484615385 X (slope) 0.003942308

  9. Exercise 8 R function ols(a,b) returns $coef EstimateStd. Error t value Pr(>|t|) (Intercept) 0.484615385 1.289275061 0.3758821 0.7199360 x 0.003942308 0.002137246 1.8445735 0.1146492 $Ftest.p.value value 0.1146492 Thismeansthat SAT accounts for about 36% of the variance in GPA. Thisgives an indication of the strength of the assocition $R.squared [1] 0.3618685

  10. Exercise 9 x=c(40,41,42,43,44,45,46) y=c(1.62,1.63,1.90,2.64,2.05,2.13,1.94) ols(x,y) $coef EstimateStd. Error t value Pr(>|t|) (Intercept) -1.25321429 2.73157319 -0.4587885 0.6656396 x 0.07535714 0.06345636 1.1875429 0.2883482 $Ftest.p.value value 0.2883482 $R.squared [1] 0.2200002

  11. Exercise 10 c=c(300,280,305,340,348,357,380,397,453,456,510,535,275,270,335,342,354,394,383,450,446,513,520,520) d=c(32.75,28,30.75,29,27,31.20,27,27,23.50,21,21.5,22.8,30.75,27.25,31,26.50,23.50,22.70,25.80,27.80,21.50,22.50,20.60,21) Ols(c,d) yields (Intercept) 39.99094634 X(slope) -0.03565283 600 exceeds the range of X values, so the prediction is based on extrapolation.The relationship between the variables may change in extreme values.

  12. Exercise 11 mou=c(63.3,60.1,53.6,58.8,67.5,62.5) time=c(241.5,249.8,246.1,232.4,237.2,238.4) R function cor.test(mou,time) returns Pearson's product-moment correlation t = -0.7872, df = 4, p-value = 0.4752 sampleestimates:cor -0.3662634 There is insufficientevidence to determinethat the correlation is differentthan 0. > qt(0.975,4): [1] 2.776445 pt(-0.7872,4): [1] 0.2375939, for two tailed 0.234*2=0.475 P>0.05 T=-0.78 doesnot Exceedcrticialvalue Of 2.77 or -2.77

  13. Exercise 12 x=c(1,2,3,4,5,6) y=c(1,4,7,7,4,1) ols(x,y) (Intercept) 4.000000e+00 (slope) -5.838669e-16 (reasonably close to 0) Data is consistent with an inverted U shape rather than with the linear model. There might be an association here that is not detected.

  14. Exercise 13 x=c(1,2,3,4,5,6) y=c(4,5,6,7,8,2) The LSR slope is still 0 even though there is a clear linear trend to the data, which is masked by a single outlier

  15. Exercise 14 The nature of the relationship between two variables can vary with the predictor value. In other words, the association between Y and X can change as a function of X values. Extrapolating beyond the data range, therefore, can be problematic, even when the association appears to be linear. In non-linear associations, the LSR line can be misleading.

  16. Exercise 15 age=c(5.2,8.8,10.5,10.6,10.4,1.8,12.7,15.6,5.8,1.9,2.2,4.8,7.9,5.2,0.9,11.8,7.9,1.5,10.6,8.5,11.1,12.8,11.3,1,14.5,11.9,8.1,13.8,15.5,9.8,11.0,14.4,11.1,5.1,4.8,4.2,6.9,13.2,9.9,12.5,13.2,8.9,10.8) cpep=c(4.8,4.1,5.2,5.5,5,3.4,3.4,4.9,5.6,3.7,3.9,4.5,4.8,4.9,3.0,4.6,4.8,5.5,4.5,5.3,4.7,6.6,5.1,3.9,5.7,5.1,5.2,3.7,4.9,4.8,4.4,5.2,5.1,4.6,3.9,5.1,5.1,6.0,4.9,4.1,4.6,4.9,5.1) R function: cor(age,cpep) returns; [1] 0.3906776 R function: hc4test(age,cpep) returns: $test [1] 4.705966 $p.value [1] 0.03005811 Thus, r=0.39, and the hc4test rejects at 0.05

  17. Exercise 16 age=c(5.2,8.8,10.5,10.6,10.4,1.8,12.7,15.6,5.8,1.9,2.2,4.8,7.9,5.2,0.9,11.8,7.9,1.5,10.6,8.5,11.1,12.8,11.3,1,14.5,11.9,8.1,13.8,15.5,9.8,11.0,14.4,11.1,5.1,4.8,4.2,6.9,13.2,9.9,12.5,13.2,8.9,10.8) cpep=c(4.8,4.1,5.2,5.5,5,3.4,3.4,4.9,5.6,3.7,3.9,4.5,4.8,4.9,3.0,4.6,4.8,5.5,4.5,5.3,4.7,6.6,5.1,3.9,5.7,5.1,5.2,3.7,4.9,4.8,4.4,5.2,5.1,4.6,3.9,5.1,5.1,6.0,4.9,4.1,4.6,4.9,5.1) ols(age[age<7],cpep[age<7]) $coef EstimateStd. Error t value Pr(>|t|) (Intercept) 3.5148814 0.37014633 9.495924 6.244186e-07 x 0.2474008 0.08924835 2.772049 1.689761e-02 C-peptide concentrations increase to about age 7. The regression line plateaus beyond that age. Using a single line or correlation To describe the relationship is misleading ls(age[age>7],cpep[age>7]) $coef EstimateStd. Error t value Pr(>|t|) (Intercept) 4.7535568 0.64125948 7.4128445 5.654828e-08 x 0.0132083 0.05550626 0.2379606 8.137083e-01

  18. Exercise 17 size=c(2359,3397,1232,2608,4870,4225,1390,2028,3700,2949,688,3147,4000,4180,3883,1937,2565,2722,4231,1488,4261,1613,2746,1550,3000,1743,2388,4522) price=c(510,690,365,592,1125,850,363,559,860,695,182,860,1050,675,859,435,555,525,805,369,930,375,670,290,715,365,610,1290) R Function ols(size,price) returns (Intercept) 38.1921217 X (Slope) 0.2153008 The conclusion hereisthat a home size of 0 cost 38.192, whichmakes no sense. This illustrates ho non-linearrelationshipscanmake the regression land midleading. Extrapolation beyond the data canbeproblematic.

  19. Exercise 18 lot=c(18200,12900,10060,14500,76670,22800,10880,10880,23090,10875,3498,42689,17790,38330,18460,17000,15710,14180,19840,9150,40511,9060,15038,5807,16000,3173,24000,16600) price=c(510,690,365,592,1125,850,363,559,860,695,182,860,1050,675,859,435,555,525,805,369,930,375,670,290,715,365,610,1290) R function ols(lot,price) returns EstimateStd. Error t value Pr(>|t|) (Intercept) 436.83367567 66.609568133 6.558122 5.927679e-07 x (slope) 0.01104288 0.002754693 4.008752 4.569549e-04

  20. Exercise 19 This would generally be the case when the relationship are linear and homoscedastic.

  21. Exercise 20 x=c(18,20,35,16,12) y=c(36,29,48,64,18) R function ols(x,y) returns: EstimateStd. Error t value Pr(>|t|) (Intercept) 25.3283679 23.774217 1.0653713 0.3648449 x 0.6768135 1.096856 0.6170485 0.5808715 $Ftest.p.value: 0.5808715 R functioncor.test(x,y) returns: t = 0.617, df = 3, p-value = 0.5809 sampleestimatescor: 0.3355929 Both analyses agree, both not significant. X and Y can still be dependent in nonlinear ways, and there are power considerations with a small sample size.

  22. Exercise 21 x=c(12.2,41,5.4,13,22.6,35.9,7.2,5.2,55,2.4,6.8,29.6,58.7) y=c(1.8,7.8,0.9,2.6,4.1,6.4,1.3,0.9,9.1,0.7,1.5,4.7,8.2) R function ols(x,y) returns EstimateStd. Error t value Pr(>|t|) (Intercept) 0.3269323 0.248122843 1.317623 2.144131e-01 x 0.1550843 0.008413901 18.431919 1.280856e-09 The estimate of the slopeis 0.155 with a SE of 0.0084. The 0.975 quantile of Twith 24 dfis: > qt(0.975,24) [1] 2.063899 The scatter plot suggests that X and Y increase together, but with the same confidence interval situations arise when it is not always the case

  23. Exercise 22 x=c(34,49,49,44,66,48,49,39,54,57,39,65,43,43,44,42,71,40,41,38,42,77,40,38,43,42,36,55,57,57,41,66,69,38,49,51,45,141,133,76,44,40,56,50,75,44,181,45,61,15,23,42,61,146,144,89,71,83,49,43,68,57,60,56,63,136,49,57,64,43,71,38,74,84,75,64,48) y=c(129,107,91,110,104,101,105,125,82,92,104,134,105,95,101,104,105,122,98,104,95,93,105,132,98,112,95,102,72,103,102,102,80,125,93,105,79,125,102,91,58,104,58,129,58,90,108,95,85,84,77,85,82,82,111,58,99,77,102,82,95,95,82,72,93,114,108,95,72,95,68,119,84,75,75,122,127) R function ols(x,y) returns $coef EstimateStd. Error t value Pr(>|t|) (Intercept) 97.95728197 4.73432147 20.6908809 9.985891e-33 x (slope) -0.02136595 0.07096758 -0.3010664 7.641969e-01 pq(0.975.df=77) [1] 1.99

  24. Exercise 23 khomreg(size,price) $test [1,] 6.115014, $p.value[1,] 0.01340384 khomreg(lot,price) $test [1,] 0.1683221 $p.value [1,] 0.6816073 We actually do reject for house size but not for lot size. This test may not have sufficient power to detect heteroscedasticity, so when we fail to reject, it is difficult to draw conclusions

  25. Exercise 24 ols(x,y) EstimateStd. Error t valuePr(>|t|) (Intercept) 65.46175413 18.4508380 3.5479014 0.000673844 x (slope) -0.05649584 0.1876524 -0.3010664 0.764196940 (Slopeis close to 0, with P<0.764, do not rejet with OLS) $Ftest.p.value value 0.7641969 (bookhastypo) rqfit(x,y) $coef (Intercept) x 95.2000000 -0.4333333 $ci lower bd upper bd (Intercept) 64.4610733 105.972735 X (Slope) -0.5505706 -0.1450298 (CI for slopedoes not contain 0, sorejectwithrqfit. ols(y,x) regplot(y,x,regfun=rqfit) As isevidentin the scatterplot of OLS, there are severaloutliersbetween the X values of 100-130. To minimize least squared distances, theseoutliers pull the regression line upward in a mannerthatmakesithorizontal.Therqfitisbased on the median of Y instead of mean. It isthusinsensitive to outliers , making the regressionline (in blue) go throughthe middle (0.5 y quantile/X) of the bulk of the observations.

  26. Exercise 25 The data can be accessed by library(MASS) X=c(2300,750,4300,2600,6000, 10500, 10000, 17000, 5400, 7000, 9400, 32000, 35000, 100000, 100000, 52000, 100000, 4400, 3000, 4000, 1500, 9000, 5300, 10000, 19000, 27000, 28000, 31000, 26000, 21000, 79000, 100000,100000)       Y=c(65,156,100,134,16,108,121,4,39,143,56,26,22,1,1,5,65,56,65,17,7,16,22,3,4,2,3,8,4,3,30,4,43) ols(X,Y) $coef EstimateStd. Error t value Pr(>|t|) (Intercept) 53.8899623928 1.027986e+01 5.242286 1.072131e-05 x -0.0004461206 2.296306e-04 -1.942775 6.117379e-02 $Ftest.p.value0.06117379 Olshc4 reject. It has a smaller standard error for the slope olshc4(X,Y) $ci Coef. Estimates ci.lowerci.upper p-value Std.Error (Intercept) 0 53.8899623928 30.5619402421 7.721798e+01 4.902827e-05 1.143803e+01 Slope 1 -0.0004461206 -0.0008776261 -1.461508e-05 4.315956e-02 2.115728e-04

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