If x and y are directly related r 0 If x and y are inversely related r 0

If x and y are directly related r 0 if x and y are

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If  x  and  y  are directly related,  r  > 0. If  x  and  y  are inversely related,  r  < 0. Strength  of the linear relationship between       x  and  y. The larger the absolute value of  r , the more the value of  y  depends in  a linear way on the value of  x .
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 4/24/11 Coefficient of Determination A measure of the Strength  of the linear relationship between  x  and  y. The larger the value of  r 2, the more the value of  y  depends in a  linear way on the value of  x . Amount of variation in  y  that is related   to variation in  x . Ratio of variation  in  y  that is explained by the  regression model divided by the total variation in  y .
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 4/24/11 Testing for Linearity Key Argument: If the value of  y  does  not  change linearly with the  value of  x , then using the mean value of  y  is the  best predictor for the actual value of  y .  This  implies            is preferable. If the value of  y  does change linearly with the  value of  x , then using the regression model gives  a better prediction for the value of y than using  the mean of  y .  This implies         is preferable. y = y y = ˆ y
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 4/24/11 Three Tests for Linearity 1. Testing the Coefficient of Correlation H0:  ρ  = 0  There is no linear relationship between  x  and   y . H1:  ρ    0  There is a linear relationship between  x  and  y . Test Statistic:  2. Testing the Slope of the Regression Line H0:  β 1  = 0  There is no linear relationship between  x  and   y . t = r 1 r 2 n 2 t b s y x x n x = - ⋅ 1 2 2 , ( )
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 4/24/11 Three Tests for Linearity 3. The Global  F -test H0: There is no linear relationship between  x  and  y . H1: There is a linear relationship between  x  and  y . Test Statistic: Note:  At the level of simple linear regression, the  global  F -test is equivalent to the  t -test on  β 1.  When we  conduct regression analysis of multiple variables, the  F = MS R MS E = SS R 1 SS E ( n 2)
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 4/24/11 Excel Output, Problem 15.9 SUMMARY OUTPUT Regression Statistics Multiple R 0.848584 R Square 0.72009481 Adjusted R Square 0.67344395 Standard Error 91.4789339 Observations 8 ANOVA df SS MS F Significance F Regression 1 129173.1279 129173.128 15.43583 0.00772299 Residual 6 50210.37209 8368.39535 Total 7 179383.5 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 44.3139535 108.5086985 0.40839079 0.69716178 -221.197461 309.825368 Years 38.755814 9.864427133 3.92884589 0.00772299 14.6184126 62.8932153 The calculated  t  for the test of H0:  β 1 = 0 The  global  F   test  statistic  for the  test of   H0:  β 1 = 0 Note that: (1) both  t  and  F  have the  same  p -value,  and  (2)  t 2 =  F.
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