LESSON 24: INFERENCES USING REGRESSION Outline Assumptions/Pitfalls in Regression Analysis Coefficient of Determination/Correlation Coefficient Inferences using the Regression Line
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
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
PITFALLS IN MULTIPLE REGRESSION Multicollinearity High correlation between two independent variables Solution: eliminate one of the two variables that are correlated
COEFFICIENT OF DETERMINATION Total sum of squares / Total variation Regression sum of squares / Explained variation Error sum of squares / Unexplained variation
COEFFICIENT OF DETERMINATION
INFERENCES USING REGRESSION LINE Confidence interval for the conditional mean using small samples
INFERENCES USING REGRESSION LINE Prediction interval for an individual Y using small samples
INFERENCES USING REGRESSION LINE Confidence interval estimate of the true slope B
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
INFERENCES USING REGRESSION LINE Confidence interval estimate of the true intercept A
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 14 68 23 105 9 40 17 79 10 51
INFERENCES USING REGRESSION LINE
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 14 68 23 105 9 40 17 79 10 51
INFERENCES USING REGRESSION LINE
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 14 68 23 105 9 40 17 79 10 51
INFERENCES USING REGRESSION LINE
READING AND EXERCISES Lesson 24 Reading: Section 12-1 to 12-3 pp. 412-433 Exercises: 12-11, 12-13, 12-19