Welcome to BUAD 310

1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 20, Monday April 7, 2014

2. Agenda & Announcement • Today: • Finish off the in class exercise from last time • Chapter 22, Inference for SRM • Homework 6 is due this Wednesday April 9 by 5 PM. • Just a reminder: • Exam II on Wednesday April 16, 33 questions. • Similar to Exam I • All the stuff (practice test, etc.) will go up on Wednesday April 9th. • NO CELL PHONES ALLOWED. BUAD 310 - Kam Hamidieh

3. Population vs. Estimated Lines The blue line:: All the black dots: The blue dots form our random sample. The red line: The residuals: BUAD 310 - Kam Hamidieh

4. Sample versus Population Unobserved estimates BUAD 310 - Kam Hamidieh

5. From Last Time • Quality of your regression: • r2 = the percentage of the response variation accounted for by the regression. • Se = The standard deviation of the residuals measures the average distance of the points from the line. • Use the residuals for • Checking normality of errors • Checking to see variance (or sd) is constant • QQ Plots = new graphical tool to check that the residuals are normal BUAD 310 - Kam Hamidieh

6. Inference Goals: • Carry out hypotheses tests to see if X and Y are linearly related in the larger population represented by the sample • Use sample regression results to get confidence intervals for the slope and Y values at a fixed X • Check the model; do our original SRM assumptions pan out? (Discuss some remedies when we do multiple regression.) BUAD 310 - Kam Hamidieh

7. Inference about the Slope • If B1 = 0, then there is no linear relationship between Y and X. X will not be useful in predicting Y values! • We do not see B1 but see b1. • We can assess if B1 = 0 by: • Conduct a hypothesis test with H0: B1 = 0 • Construct a confidence interval for B1 and see if zero is in it BUAD 310 - Kam Hamidieh

8. Doing the Usual… Hypothesis Tests: Confidence Intervals: BUAD 310 - Kam Hamidieh

9. Standard Error of b1 • The quantity b1 is a random variable. • It has its own sampling distribution. (What is a sampling distribution? But you remember!) • The standard deviation of the b1is estimated as follows: • Does the formula make sense? BUAD 310 - Kam Hamidieh

10. CI and Tests for B1 To test H0: B1 = 0 vs. Ha: B1 ≠ 0: (1) 100(1-α)% confidence interval for B1 is: b1 ± tα/2se(b1) where tα/2comes from a t distribution with df = n-2. Or (2) Compute the test statistics: then get the p-value from a t distribution with df = n-2. BUAD 310 - Kam Hamidieh

11. CI and Tests for B0 • The same idea holds for B0. • However, you usually don’t test B0 = 0. • We will let software get us all the standard errors but do include the formulas in your cheat sheet. BUAD 310 - Kam Hamidieh

12. Apple vs. SP500 From: finance.yahoo.com, daily % changes, 5/31/2013 to 10/21/2013 BUAD 310 - Kam Hamidieh

13. Apple Regression Results 95% CI for B1: (0.057, 0.873) How about: H0: B1 = 0 Ha: B1 ≠ 0 Test Stat? P-Value? BUAD 310 - Kam Hamidieh

14. Apple Regression Results Test: H0: B1 = 0 Ha: B1 ≠ 0 Test Stat ≈ 2.26 P-Value ≈ 0.0259 BUAD 310 - Kam Hamidieh

15. In Class Exercise 1 Refer to our In Class Exercise 1 from Lecture 19. What is the value of the test statistics and p-value for testing: (Hint: slide 10)H0: B1 = 0 vs Ha: B1 ≠ 0 What is the 95% confidence interval for B1? (Hint: look at the output!) Is there a statistically significant linear relationship between price and score? Why? Suppose you want to test at α=0.05:H0: B1 = 0.06 vs Ha: B1 ≠ 0.06What would be your conclusion? (Hint: you need not compute a p-value!) A non statistician makes the following comment: On averagecamera scores could go up by 3 points when you spend an additional $50. Agree/Disagree? Why? (Hint: look the range implied by the 95% confidence interval & the interpretation of slope.) BUAD 310 - Kam Hamidieh

16. New Y and Mean Y There are two types of intervals used to make inference about the response variable Y for given x: • Confidence intervals for mean response • Prediction for a future unobserved value (Cover in multiple regression) BUAD 310 - Kam Hamidieh

17. CI for Mean Response • Recall that: • After least squares we get: • b0, b1 are random variables and have uncertainty associated with them. Then so does • CI for : gives a range of reasonable values for the mean of Y’s at a fixed x. BUAD 310 - Kam Hamidieh

18. CI for Mean Response 100(1-α)% confidence interval for at xnew is: ) where tα/2comes from a t distribution with df = n-2, and BUAD 310 - Kam Hamidieh

19. Apple vs S&P 500 (How?) Why do the green lines narrow and then widen? BUAD 310 - Kam Hamidieh

20. Example For Apple vs. S&P 500, we have: What is the 95% confidence interval for mean Apple return when S&P 500 goes up by 1%? From StatCrunch: xnew = 0.01, x-bar = 0.00053, SX = 0.0076, n = 100, Se = 0.0156 BUAD 310 - Kam Hamidieh

21. Example Continued 95% Confidence interval gives: ) = 0.00595 ± (1.98?)(0.0025) = = (0.001, 0.011) When S&P 500 goes up by 1%, we are 95% confident that the mean Apple return will be between 0.1% and 1.1%. BUAD 310 - Kam Hamidieh

22. In Class Exercise 2 This will be handed out in class. BUAD 310 - Kam Hamidieh

23. Next Time • Continue with regression. BUAD 310 - Kam Hamidieh