Lecture+17+su10+_regression+HT_ - Sociology 210 Lecture 17...

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Sociology 210 Lecture 17: Hypothesis Testing in Bivariate Regression
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02/23/11 SOC 210 summer 2010 2 The Least Squares Solution The estimates of a and b that give the smallest possible value of Σ e 2 (i.e. that minimize the sum of the squared residuals) are: One nice feature of these estimates is that the line will always pass through the point at the mean of X and mean of Y. Another is that the mean of Y and the mean of Yhat will always be the same. X b Y a and SS SP X X Y Y X X b X XY - = = - - - = ) ( ) )( ( 2
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02/23/11 SOC 210 summer 2010 3 Interpreting the Estimates An estimated intercept ( a ) of 3.09 tells us that when X = 0 (i.e. for sophomores), the expected work hours per week ( Y ) 3.09. An estimated slope ( b ) of 6.58, tells us that each year ( X ) the number of work hours ( Y ) increases by 6.58 hours. X bX a Y 58 . 6 09 . 3 ˆ + = + =
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02/23/11 Soc 210 summer 2010 4 Goodness of Fit To assess how well our model fits the data, we can look at the ratio of the Model SS to the Total SS. This is called R 2 : R 2 measures the proportion of the total variability in Y that is accounted for by the model. It varies from 0 to 1. It turns out that R 2 is the square of the correlation of X and Y. TSS RSS TSS MSS R - = = 1 2
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02/23/11 Soc 210 summer 2010 5 Goodness of Fit in Example TSS RSS TSS MSS R Y Y RSS Y Y MSS Y Y TSS i i i i - = - = = = = = - = = - = = - = 1 1368 92 . 847 1 38 . 0 1368 08 . 520 92 . 847 ) ˆ ( 08 . 520 ) ˆ ˆ ( 1368 ) ( 2 2 2 2 About 38% of the variation in work hours (Y) is accounted for by year in school (X)
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Conceptual Review of Hypothesis Testing Our estimate (mean, proportion, coefficient, etc) is based on a sample from a population Sampling error leads to differences across samples in the estimate We want to know whether the difference between our estimate and the estimate that would be expected under the null hypothesis is merely due to sampling error. In other words, how unusual is our estimate compared to what we would expect under the null hypothesis? Hypothesis testing compares our sample estimate to the distribution of sample estimates expected under the null hypothesis To do this, we must use the standard error of the estimate (the standard deviation of the coefficient estimates across samples).
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