1. Inference for Regression: Inference About the Model Section 14.1

2. Review LSRL Example: Natural gas Consumption. 1.Make a scatterplot of data. 2.Calculate the LSRL. 3.Look for outliers and influential observations. 4.Look at correlation coefficient r and r 2.

3. The Regression Model: Slope = b and Intercept = a Both a and b are statistics that predict the true parameters α and β of the true regression equation. Assumptions: –For any given x, y varies normally. –The mean response, µ y, has a linear relationship with x: µ y = α + βx –α = intercept and β = slope and both are unknown parameters. –St. dev. of y (σ) is the same for all values of x and σ is unknown.

4. Regression Model

5. Inference About the Model First find estimators of the unknown parameters α, β, σ. Use LSRL: y =a + bx –b is unbiased estimator of true slope β. –a is unbiased estimator of true intercept α. –σ requires some calculations. σ describes the variability of y about the LSRL so we use residuals to estimate σ. –Residual = observed y – predicted y = y – y hat

6. Sample Standard Deviation: Standard Error: s = 1/n-2 (y-yhat) 2 Degrees of freedom = n-2 (2 variables). Calculator: –L1, x values. –L2, y values. –L RES, L2 – Y1(L1) gives residuals. –1-var stats L RES gives x 2 = residuals 2 needed for the formula for s.

7. Confidence Intervals for regression Slope, β Estimate β using b. b ± t* SE b t* from Confidence Level with n-2 df. SE b = s/(x – x) 2 SE b is calculated for you. Example: Calculate a 95% confidence interval for the average gas use per degree day.

8. Example: Gas use per degree day b ± t* SE b.1890 ± 2.145(.004934) (.1784,.1996)

9. Hypothesis Testing of No Linear Relationship H 0 : β = 0 (no linear relationship or no correlation between x and y) H a : β, or 0. Test statistic: t = b/SE b Calculator: LinRegTTest

10. Example: Golfer Golfer 123456789101112 Round 1 89908795868110210583889179 Round 2 948589 81761078987918880 How well do golfers first round scores predict their second round scores? Linear? LSRL HATS