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Unformatted text preview: andard deviation that does not depend on x. This true mean is sometimes expressed as E(Y) = 0 + 1(x). And the components and assumptions regarding this statistical model are shown visually below. The represents the true error term. These would be the deviations of a particular value of the response y from the true regression line. As these are the deviations from the mean, then these error terms should have a normal distribution with mean 0 and constant standard deviation . Now, we cannot observe these ’s. However we will be able to use the estimated (observable) errors, namely the residuals, to come up with an estimate of the standard deviation and to check the conditions about the true errors. 193 So what have we done, and where are we going? 1. Estimate the regression line based on some data. DONE! 2. Measure the strength of the linear relationship with the correlation. DONE! 3. Use the estimated equation for predictions. DONE! 4. Assess if the linear relationship is statistically significant. 5. Provide interval estimates (confidence intervals) for our predictions. 6. Understand and check the assumptions of our model. We have already discussed the descriptive goals of 1, 2, and 3. For the inferential goals of 4 and 5, we will need an estimate of the unknown standard deviation in regression Estimating the Standard Deviation for Regression The standard deviation for regression can be thought of as measuring the average size of the residuals. A relatively small standard deviation from the regression line indicates that individual data points generally fall close to the line, so predictions based on the line will be close to the actual values. It seems reasonable that our estimate of this average size of the residuals be based on the residuals using the sum of squared residuals and dividing by appropriate degrees of freedom. Our estimate of is given by: s= sum of squared residuals n2 SSE MSE where SSE n2 e 2
i ˆ2 y y Note: Why n – 2? In estimating the mean respo...
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