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3. Recall our Regression Example Exam 2 vs Final Least Squares Regression Line: r 2 = 0.791 r = 0.889 Exam 2336544646040 Final538078938858 On to Inference: Sample reg line vs Population reg line

4. On to Inference: Sample versus Population pg 192 Regression Line for the Sample From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

5. On to Inference: Sample versus Population Regression Line for the Population From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

6. Inference in Linear Regression For each x, the population of y values are normally distributed with some mean (may depend on x in linear way) and a std deviation s that does not depend on x Linear Model: Response y = [b 0 + b 1 (x)] +  = [Population relationship] + Randomness From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012. Used with permission.

7. Inference in Linear Regression For each x, the population of y values are normally distributed with some mean (may depend on x in linear way) and a std deviation s that does not depend on x

8. Inference in Linear Regression  ’s = true error terms (not observe), and have normal distribution with mean 0 and std deviation . We cannot see  ’s --- but can see residuals (observed errors); so use residuals to assess if all ok about true error assumptions.

9. Goals in Regression: pg 194 1.Estimate regression line based on data. 2.Measure strength of the linear relationship with the correlation. 3.Use estimated equation for predictions. 4.Assess if the linear relationship is statistically significant. 5.Provide interval estimates (CIs) for our predictions. 6.Understand and check the assumptions of our model.

10. Estimating Std Dev for Regression Measuring the average size of the residuals. s = Note: Why n – 2?

11. Estimating the Standard Deviation: Exam 2 and Final Exam Scores

12. Significant Linear Relationship? (pg 195) H 0 :  1 = 0 versus H a :  1 ≠ 0 What happens if the null hypothesis is true?

13. t-test for the population slope  1 To test H 0 :  1 = 0 we would use where and degrees of freedom for t-distribution are n – 2. Could be modified to test a variety of hypotheses.

14. Try It! Significant Linear Relationship between Exam 2 and Final Scores? Is there a significant (non-zero) linear relationship between exam 2 score and final exam score? Is exam 2 a useful linear predictor for final score? Test H 0 :  1 = 0 versus H a :  1 ≠ 0 at the 5% level.

15. A = Yes or B = No pg 196 Based on previous t-test at 5% significance level, do you think a 95% confidence interval for true slope would contain the value of 0?

16. Exam 2 and Final Exam Scores Compute the 95% CI for the population slope Could you interpret the 95% confidence level.? Confidence Interval for population slope  1 where df = n-2 for the t* value

17. Inference about the population slope using SPSS

18. SPSS ANOVA F-test for Regression Note: Third way to test H 0 :  1 = 0 versus H a :  1 ≠ 0

19. Recap pages 195-196 Learning about the popul slope  1 1. T-test for  1 … df = n – 2 2. CI for  1 … df = n – 2 3. F-test for  1 … df = 1, n – 2

20. Which of the following could be used to test H 0 :  1 = 0 vs H a :  1 ≠ 0? Select all that apply. A) t-test B) CI C) F-test

21. Which of the following could be used to test H 0 :  1 = 2 vs H a :  1 ≠ 2? Select all that apply. A) t-test B) CI C) F-test

22. Which of the following could be used to test H 0 :  1 = 0 vs H a :  1 > 0? Select all that apply. A) t-test B) CI C) F-test

23. Predicting for Individuals versus Estimating the Mean How would you predict the final exam score for Barb who scored 60 points on exam 2? How would you estimate the mean final exam score for all students who scored 60 points on exam 2?  estimate for predicting a future observation and for estimating the mean response are same. What about their standard errors?

24. Predicting for Individuals versus Estimating the Mean A population of individuals and a population of means… Std dev for a population of individuals? Std dev for a population of means? Which standard deviation is larger? So a prediction interval for an individual response will be (wider or narrower) than a confidence interval for a mean response.

25. Predicting for Individuals versus Estimating the Mean

26. Try It! Exam 2 versus Final Exam Construct a 95% CI for mean final exam score for all students who scored x = 60 points on exam 2.

27. Try It! Exam 2 versus Final Exam Construct a 95% PI for the final exam score for a student who scored x = 60 points on exam 2.