Chapter 14 Inference for Regression

1. Chapter 14Inference for Regression AP Statistics 14.1 – Inference about the Model 14.2 – Predictions and Conditions

2. Two Quantitative Variables • Plot and Interpret • Explanatory Variable and Response Variable • FSDD • Numerical Summary • Correlation (r) – describes strength and direction • Mathematical Model • LSRL for predicting

3. Conditions for Regression Inference • For any fixed value of x, the response y varies according to a Normal distribution • Repeated responses y are Independent of each other • Parameters of Interest: • The standard deviation of y (call it ) is the same for all values of x. The value of is unknown  t-procedures! • Degrees of Freedom: n – 2

4. Conditions for Regression Inference (Cont’d) • Look at residuals: • residual = Actual – Predicted • The true relationship is linear • Response varies Normally about the True regression line • To estimate , use standard error about the line (s)

5. Inference • Unknown parameters: • a and b are unbiased estimators of the least squares regression line for the true intercept and slope , respectively • There are n residuals, one for each data point. The residuals from a LSRL always have mean zero. This simplifies their standard error.

6. Standard Error about the Line • Two variables gives: n – 2 df (not n – 1) • Call the sample standard deviation (s) a standard error to emphasize that it is estimated from data • Calculator will calculate s! Thank you TI!

7. t-procedures (n - 2 df) • CI’s for the regression slope standard error of the LSRL slopeb is: • Testing hypothesis of No linear relationship • x does not predict y r = 0

8. What is the equation of the LSRL? • Estimate the parameters • In your opinion, is the LSRL an appropriate model for the data? Would you be willing to predict a students height, if you knew that his arm span is 76 inches?

11. Construct a 95% CI for mean • increase in IQ for each additional • peak in crying

13. Scatter Plot and LSRL?Perform a Test of Significance

17. Checking the Regression Conditions • All observations are Independent • There is a true LINEAR relationship • The Standard Deviation of the response variable (y) about the true line is the Same everywhere • The response (y) varies Normally about the true regression line * Verifying Conditions uses the Residuals!