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Class_19 - Significance Tests for Regression Models and...

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Significance Tests for Regression Models and Their Coefficients
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A. Testing the Significance of the Regression Coefficient The null hypothesis in the significance test for the regression coefficient (i.e., slope) is: H 0 : β = 0.0 This simple symbolic expression says more than might first appear. It says: If we begin by assuming that there is no relationship between X and Y in general (i.e., in the universe from which our sample data come), then how likely is it that we would find a regression coefficient for our sample to be DIFFERENT FROM 0.0? Put the other way around, if we find a relationship between X and Y in the sample data, can we infer that there is a relationship between X and Y in general?
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To test this null hypothesis, we use our old friend the t-test: where the standard error is: b b t σ β ˆ - = ( 29 1 ˆ 2 - = N s MS X Error b
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This is the standard deviation of the sampling distribution of all theoretically possible regression coefficients for samples of the same size drawn randomly from the same universe. Recall that the mean of this sampling distribution has a value equal to the population characteristic (parameter), in this case the value of the regression coefficient in the universe. Under the null hypothesis, we initially assume that this value is 0.0. To test the significance of the regression coefficient (and the model as a whole), we need the statistical information found in the usual analysis of variance summary table. We already have most of this information for our previous example.
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Recall that R 2 YX , the Coefficient of Determination, was found from R 2 YX = SS Regression / SS Total From our time/temperature example, remember that R 2 YX was 0.999. Total sum of squares can be found from SS Total = s Y 2 (N - 1) From our previous calculations, remember that s Y 2 was 49.333. Thus, SS Total is SS Total = (49.333) (3 - 1) SS Total = (49.333) (2) SS Total = 98.667
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By rearranging the algorithm for R 2 YX , we get SS
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Class_19 - Significance Tests for Regression Models and...

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