1. LESSON 24: INFERENCES USING REGRESSION
Outline
Assumptions/Pitfalls in Regression Analysis
Coefficient of Determination/Correlation Coefficient
Inferences using the Regression Line

2. LINEAR REGRESSION ASSUMPTIONS
The conditional means all lie on the same straight line
and the sample observations
All populations have the same standard deviation
Successive sample observations are independent
The value of is known in advance

3. LINEAR REGRESSION ASSUMPTIONS
The true regression coefficients and are usually unknown and are estimated from the sample counterparts
For multiple regression, the true regression plane
and each sample observation

4. PITFALLS IN MULTIPLE REGRESSION
Multicollinearity
High correlation between two independent variables
Solution: eliminate one of the two variables that are correlated

5. COEFFICIENT OF DETERMINATION
Total sum of squares / Total variation
Regression sum of squares / Explained variation
Error sum of squares / Unexplained variation

6. COEFFICIENT OF DETERMINATION

7. INFERENCES USING REGRESSION LINE
Confidence interval for the conditional mean using small samples

8. INFERENCES USING REGRESSION LINE
Prediction interval for an individual Y using small samples

9. INFERENCES USING REGRESSION LINE
Confidence interval estimate of the true slope B

10. INFERENCES USING REGRESSION LINE
Hypotheses Test regarding B: A testing procedure will find whether or not two variables are correlated, and if so whether the slope is positive or negative
Hypotheses:
Test Statistics:
d.f.=n - 2

11. INFERENCES USING REGRESSION LINE
Confidence interval estimate of the true intercept A

12. INFERENCES USING REGRESSION LINE
Example 1: Refer to Lesson 20, Example 1 (rewritten below). Construct a 98% confidence interval estimate of the conditional mean final product weight when the raw-materials volume is 25 gallons. Recall that Y = weight of final product in pounds and X = volume of raw materials in gallons.
X Y

13. INFERENCES USING REGRESSION LINE

14. INFERENCES USING REGRESSION LINE
Example 2: Refer to Example 1 (rewritten below). Construct a 98% prediction interval estimate of the final product weight in an individual batch having 20 gallons of raw materials. Recall that Y = weight of final product in pounds and X = volume of raw materials in gallons.
X Y

15. INFERENCES USING REGRESSION LINE

16. INFERENCES USING REGRESSION LINE
Example 3: Refer to Example 1.
Construct a 95% confidence interval estimate of the slope of the true regression line.
At the 5% significance level, must you accept or reject the null hypothesis that final product weight (Y) and raw materials volume (X) are uncorrelated variables?
X Y

17. INFERENCES USING REGRESSION LINE

18. READING AND EXERCISES Lesson 24 Reading:
Section 12-1 to 12-3 pp
Exercises:
12-11, 12-13, 12-19