Transcript Chapter 11
Chapter 14 Inference for Regression AP Statistics 14.1 – Inference about the Model 14.2 – Predictions and Conditions 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 yˆ a bx 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: y x • 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 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) 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. Standard Error about the Line • Two variables gives: n – 2 df (not n – 1) 1 2 s residual n2 • Call the sample standard deviation (s) a standard error to emphasize that it is estimated from data • Calculator will calculate s! Thank you TI! t-procedures (n - 2 df) • CI’s for the regression slope standard error of the LSRL slope b is: s b SE b 2 ( x x ) • Testing hypothesis of No linear relationship b t * SE H0 : 0 b t statistic : t SEb • x does not predict y r = 0 • What is the equation of the LSRL? • Estimate the parameters and • 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? •Construct a 95% CI for mean increase in IQ for each additional peak in crying Scatter Plot and LSRL? Perform a Test of Significance 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!