Linear Regression Problems

1. Chapter 13 Linear Regression Problems Solved Problem

2. Problem 13-1 (1)Note: this data applies to problems 13-1, 13-13, 13-21,13-27 Right Click here and select “open hyper link” for excel solution1 excel solution2

3. Problem 13-1 (2)

4. Problem 13-1 r = .75 , Is this statistically significant? no relationship Note: We should not proceed further with this regression model However the text does and it provides good examples of the calculators used in later problems.

5. Problem 13-2 (1)Note: this data applies to problems 13-2, 13-14, 13-22,13-28 Right click here and “Open hyper link” for the excel sol. Click here for the excel sol2.

6. Problem 13-2

7. Problem 13-2

8. Problems 13-2, 13-14,13-22, 13-28Excel Solution Note: this will be -.8908 since the sign of the correlation coefficient, r must agree with the slope of the regression equation b.

9. Problems 13-2, 13-14,13-22, 13-28Excel Solutions Cont’d

10. Problems 13-5, 19, 25 • Police is the independent variable and crime is the dependent variable • Scatter Diagram • Find r, the correlation coefficient Use for Syx (not in text)

11. Problems 13-5, 19, 25 Cont’d a high association with an inverse relationship. Test for statistical significance – see problems 13-7 and 13-8 for examples • H0 : ρ ≥ 0 • H1 : ρ < 0 • Test Statistics – r – t • α = .05 • Decision Rule: Fail to reject H0 if t > -1.943 • Statistical Decisions : Reject H0 • Management Conclusion: There is a statistically significant relationship between number of crimes and number of police.

12. Problems 13-5, 19, 25 Cont’d • Find r2, the coefficient of determination. r2 = (0.874)2 = 0.7638  76.38% of the number of crimes is explained by the number of police. • Strong inverse relationship. As the number of police increase, the crimes decrease. • Find the regression equation. • = a + bx=29.3877-.9596x • = 29.3877 – 0.9596(20) • For each policeman added, crime goes down by approximately one unit. 25.Find the standard error of estimate SYX Computational formula (not in text) – see last two columns of original data. Syx= = Syx= = =

13. Problems 13-5, 19, 25 Cont’d Find the 95% prediction interval for Y when X = 20 police - I requested this P.I = ± t Syx = 10.1957 ± (2.447) (3.3786) = 10.1957 ± 8.267 = 10.1957 ± 8.267 = 10.1957 ± 8.8178 1.3779 < Y< 19.0135 - I am 95% sure that an estimate for an individual value of Y lies between 1.38 and 19.01 crimes, but only when X = 20 policemen.

15. Problem 13-7

16. Problem 13-8

17. Problem 13-13 (Data from #13-1)

18. a. Problem 13-13 (13-1) b. ŷ= 3.7671 + .3631(7) ŷ= 6.308 when x = 7

19. Problem 13-14 Note: this data applies to problems 13-2, 13-14, 13-22,13-28 Right click here and “Open hyper link” for the excel sol. Click here for the excel sol2.

20. Problem 13-14 (13-2) a. b. ŷ= 19.1197 + 1.7425(7) ŷ= 6.922 when x = 7

21. Problem 13-15 • X Y • 12 9 2.9 1.6 8.41 2.56 4.64 • 9 7 -0.1 -0.4 0.01 0.16 0.04 • 14 10 4.9 2.6 24.01 6.76 12.74 • 6 5 -3.1 -2.4 9.61 5.76 7.44 • 10 8 0.9 0.6 0.81 0.36 0.54 • 8 6 -1.1 -1.4 1.21 1.96 1.54 • 10 8 0.9 0.6 0.81 0.36 0.54 • 10 10 0.9 2.6 0.81 6.76 2.34 • 5 4.0 -4.1 -3.4 16.81 11.56 13.94 • 7 7.0 -2.1 -0.4 4.41 0.16 0.84 • 91 74 66.90 36.40 44.60 b. = 1.33333 + 0.66667(6) = 5.335

22. Problem 13-16

23. Problem 13-16 cont’d.. = –12.20145 + 2.19465X b. 75.5846, found by = -12.20145 + 2.19465(40)

24. Problem 13-19 • a. • b. 10.1957 found by 29.3877 – 0.9596(20) c. For each policeman added, crime goes down by one.

25. Problem 13-21 (13-1 data)Data From 13-1

26. Problem 13-21(13-1)

27. Problem 13-22 (Data from 13-2)

28. Problem 13-22(13-2)(13-2 data)

29. Problem 13-27 (13-1)

30. Problem 13-27 (13-1)

31. Problem 13-28 (13-2)

32. Problem 13-28 (13-2)