1. Chapter 27 Inferences for Regression This is just for one sample We want to talk about the relation between waist size and %body fat for the complete population. For 38 inches

2. Inferences for Regression (cont.) So the idealized regression line will have the parameters We are predicting the mean %body fat for each waist size. For each data point (x,y) we denote the error by

3. Inferences for Regression (cont.) We can also talk about each individual y as We will estimate the  parameters by finding the regression line, and the residuals “e” are the sample-base versions of the errors . Now our challenge is to account for our uncertainty in how well this parameters, calculated from a random sample, will estimate the true population parameters

4. Assumptions and Conditions Linearity (Straight Enough Condition) Check that the scatterplot of y against x has a linear form and that the scatterplot of residuals has no obvious pattern. Independent Residuals The residuals must be mutually independent. Check randomization condition Check the residual plot for evidence of patterns (which will suggest failure of independence)

5. Assumptions and Conditions (cont.) Constant Variance Check that the scatterplot shows consistent spread across the range of the x-variable, and that the residuals plot has constant variance too. Normality of the residuals Check the histogram of the residuals to see if at each x value the distribution of the residuals is nearly normal

6. Standard Error for the Slope Three aspects of the scatterplot affect the standard error of the regression slope: Spread around the line Se The spread around the line is measured with the standard deviation of the residuals

7. Standard Error for the Slope (cont.) Spread of x Values Sample Size n

8. Standard Error for the Slope A Sampling distribution for regression slopes When conditions are met, the standardized estimated regression slope, follows a student’s t-model with n-2 degrees of freedom. We estimate the standard error with Where n is the number of data values, and Sx is the ordinary standard deviation of the x-value

9. More Regression Inference Standard Error for the Intercept Hypothesis test To test hypothesis we usually choose for the null hypothesis:

10. Standard Errors for Predicted values All men with waist size x 38 inches A particular man with waist size 38 inches If we want to predict the % body fat of:

11. Standard Errors for Predicted values (cont.) Depends on : More Spread: Less accurate Prediction Less accurate slope prediction More Data: More accurate prediction Further from the center of our data: Less accurate prediction

12. Standard Errors for Predicted values (cont.) Individual values vary more than means, (so the standard error for a single predicted value has to be greater than the standard error for the mean