10.1 Simple Linear Regression

Transcription

1 10.1 Simple Linear Regression Ulrich Hoensch Tuesday, December 1, 2009

2 The Simple Linear Regression Model We have two quantitative random variables X (the explanatory variable) and Y (the response variable). The Simple Linear Regression Model assumes that where R N(0, σ). Y = β 0 + β 1 X + R, β 0, β 1, σ are the parameters of the model and must be estimated from given data of the form (x 1, y 1 ),..., (x n, y n ). An estimator for β 0 is the intercept b 0 of the regression line ŷ = b 0 + b 1 x; an estimator for β 1 is the slope b 1 of the regression line. The value of σ can be estimated using the standard deviation s of the residuals, which is computed as follows: (yi ŷ i ) 2 s = n 2.

3 The Simple Linear Regression Model

4 Example: Beer and Blood Alcohol Sixteen student volunteers at Ohio State University drank a randomly assigned number of 12-ounce cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC). Here are the data: Student x: beers y: BAC Student x: beers y: BAC

5 Example: Beer and Blood Alcohol We conduct a Linear Regression t-test on a TI calculator and interpret the results. Step 1: Press STAT, select 1: Edit... and enter the data. Step 2: Press STAT, arrow over to TESTS, and select LinRegTTest....

6 Example: Beer and Blood Alcohol Step 3: Enter the list that contains the data for the explanatory variable (X ) and the list for the response variable (Y ); set Freq: 1. As the alternative hypothesis, we will always select β & ρ: 0 (explained later); ignore RegEQ. Step 4: Press Calculate; the following screens appear.

7 Example: Beer and Blood Alcohol Interpretation of Results. Since β 0 = a = and β 1 = b = , the regression line is ŷ = x. Since the p-value is , the null hypothesis H 0 : β 1 = ρ = 0 is rejected. So, there is a significant correlation between BAC and number of beers drunk. In other words, the number of beers is a significant factor for BAC. The correlation coefficient is r = 0.894, so there is a strong positive correlation, and the coefficient of determination is % (80% of variation in BAC is explained by number of beers).

8 Example: Beer and Blood Alcohol The theoretical model is BAC = (number of beers) + R, where R N(0, ). Application. John has drunk 5 beers. According to the model above, what is his predicted BAC (after 30 minutes), and what is the probability that his BAC will be below the legal limit of 0.08? The predicted value is BAC = = P(BAC < 0.08) = normalcdf( , 0.08, , ) = 55.6%.

9 Multiple Regression and Excel Project 3 Ulrich Hoensch Thursday, April 23, 2009

10 Description of MY Project (YOURS must be different) Collect the selling prices (including shipping) for 81 TI-84 Plus Graphing calculators sold on ebay over the course of one week. The response variable is Total Price, the selling price including shipping. The explanatory variables are: Color: either BLACK (coded as 0) or SILVER (coded as 1). Condition: USED (coded as 0); NEW (coded as 1); NIB (new in box, coded as 2). Seller Score: scaled using a base-10 logarithm of actual score to reduce differences in magnitude. Seller Feedback: 1 = 100%. The model is: Total Price = β 0 + β 1 Code(Color) + β 2 Code(Condition) +β 3 LOG10(Seller Score) + β 4 Seller Feedback.

11 Excel Data Sheet

12 Coded Data Sheet

13 Regression Analysis To perform the regression analysis, follow these steps: 1. Click on Data; then click on Data Analysis. (This is Tools, Data Analysis... for the version of Excel.) 2. Select Regression, and click OK. 3. Enter the range for the response variable, and the range for the explanatory variables (preferably include labels). Set the confidence level to 95%.

14 Regression Analysis

15 Regression Analysis The output of the regression analysis is this.

16 Interpretation of Results The multiple regression coefficient is r = 0.724, and the coefficient of determination is r 2 = 0.524, so the model explains only about 50% of the variation in the selling price.

17 Interpretation of Results The coefficients are: Interpretation: The coefficient for Code(Color) is β 1 = 8.19 and the interpretation is that we can expect to pay $8.19 more for a silver calculator than for a black calculator. The coefficient for Code(Condition) is β 2 = 8.46 and the interpretation is that we can expect to pay $8.46 more for a new calculator than for a used calculator, and also $8.46 more for a new-in-box calculator than for a new calculator.

18 Interpretation of Results The coefficients are: Interpretation: The coefficient for LOG10(Seller Score) is β 3 = 2.87 and the interpretation is that we can expect to pay $2.87 more for a ten-fold increase in a seller s score. The coefficient for Seller Feedback is β 4 = and the interpretation is that we can expect to pay $2.56 more for a 1% increase in a seller s feedback score.

19 Interpretation of Results At the 95% confidence level, all variables are associated with a significant increase in the selling price (both endpoints of the confidence interval are positive), except for the seller feedback (confidence interval contains zero).