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Unformatted text preview: dividual student who scored x = 60 points on exam 2. s.e.(pred) s 2 s.e.(fit ) 2 (8.24761) 2 4.147 9.23
y t *s.e.(pred) 84.43 2.78(9.23) 84.43 25.66
=> ( 58.77, 110.09)
IT IS WIDER (and even includes
values above the max possible of 100)! Show prediction interval and
bands on the scatterplot 198 Checking Assumptions in Regression Let’s recall the statistical way of expressing the underlying model that produces our data: Linear Model: the response y = [0 + 1(x)] + = [Population relationship] + Randomness where the ‘s, the true error terms should be normally distributed with mean 0 and constant standard deviation , and this randomness is independent from one case to another. Thus there are four essential technical assumptions required for inference in linear regression: (1) Relationship is in fact linear. (2) Errors should be normally distributed. (3) Errors should have constant variance. (4) Errors should not display obvious ‘patterns’. 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. So how can we check these assumptions with our data and estimated model? (1) Relationship is in fact linear. examine the scatterplot of y versus x (2) Errors should be normally distributed. Histogram or qq plot of residuals (3) Errors should have constant variance. Residual plot (plot residuals against x);
(4) Errors should not display obvious ‘patterns’. if random scatter with no pattern in
horizontal band => ok If we saw … Let's turn to one last full regression problem that will include checking of the assumptions. 199 Relationship between height and foot length for College Men The heights (in inches) and foot lengths (in centimeters) of 32 college men were used to develop a model for the relationship between height and foot length. The scatterplot and SPSS regression output are provided. Comment on scatterplot here! D...
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