1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Eleven Part 2 (Sections 11.2 & 11.3) Chi-Square and F Distributions

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Chi Square: Goodness of Fit a test to determine if a given population follows a specified distribution

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 To Test for Goodness of Fit The Null Hypothesis, H 0 : The population fits the given distribution. The Alternate Hypothesis, H 1 : The population has a different distribution.

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Last year college students indicated that the following were their most important concerns:

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 We wish to test (at 5% level of significance) if responses for this year’s students fit last year’s distribution of percentages. We will use the Chi-square distribution.

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Chi Square

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 We wish to test (at 5% level of significance) if responses for this year’s students fit last year’s distribution of percentages. H 0 : The present distribution is the same as last year’s. H 1 : The present distribution is different.

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Suppose 400 current students were surveyed with the following results:

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Observed and Expected Frequencies

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Observed and Expected Frequencies

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Degrees of Freedom for Goodness-of-Fit Test d.f. = (number of E entries) – 1

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Degrees of Freedom = (number of E entries) – 1 = 3

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 To test the hypothesis (at 5% level of significance) that this year’s distribution is the same as last year’s Use Table 7 to find the critical value of  2 for d.f. = 3.

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Using Table 7 where d.f. = 3 and  = 0.05 The critical value of  2 = 7.81.

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Since our test statistic  2 = 23.37 is greater than the critical value  2.05 = 7.81 Reject the null hypothesis that this year’s distribution is the same as last year’s. Conclude that there has been a change in the concerns that students display.

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 P Value Approach For 3 degrees of freedom, our test statistic  2 = 23.37 is greater than the largest  2 value (12.84). Since the alpha values decrease as we move to the right, we conclude that P is less than 0.005. We would reject H 0 for any   0.005. We, therefore reject H 0 for  = 0.05.

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Testing a Single Variance or Standard Deviation Allows us to make decisions about variability.

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Reminder: In Table 7, the Area in the Right Tail of the Distribution =   22

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 If we want the  2 value for an area in the left tail of the distribution, subtract the area from one to find . 1 – left tail area =   2 = ? left tail area

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Find the  2 value such that the area to the left of  2 is 0.01, when d.f. = 6. 1 –.01 =.99 =   2 = ?.01

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 With d.f. = 6 and  = 0.990, use Table 7 to find  2.  2 = 0.872

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 If a sample of n items from a normal population (with variance  2 ) has variance s 2, then the following has a chi- square distribution with d.f. = n – 1.

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 The state law enforcement agency wishes to determine (at 1% level of significance) if a new higher speed limit has decreased the variance of speeds on a particular stretch of highway. The previous variance was 25.

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 H 0 :  2 = 25 H 1 :  2 < 25 Use  = 0.01.... has decreased the variance of speeds on a particular stretch of highway. The previous variance was 25.

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 The speeds of a sample of 30 vehicles traveling on the highway had a standard deviation of 4 m.p.h. n = 30 s = 4 so s 2 = 16 (observed)  2 = 25 (from the null hypothesis)

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Calculate  2

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 For a left-tailed test with 1% level of significance, use  = 0.99 to find the critical value of  2. 1 –.01 =.99 =   2 = ?.01

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Using Table 7 to find the critical value of  2 d.f. = 30 – 1 = 29.  = 0.99 Critical value of  2 = 14.26.

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 The observed value of  2 = 18.56. 14.26 Critical region Observed value of  2 = 18.56

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 The test statistic  2 = 18.56 does not fall within the critical region. 14.26 18.56 Critical region Do not reject the null hypothesis.

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Conclusion We cannot reject H 0. We cannot conclude that the new speed limits have resulted in lower variances.

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Estimating  2 Determining Confidence Intervals for  2

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Finding a c Confidence Interval for Variance A sample of size n is chosen from a normal population with standard deviation .

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 Chi-square Values

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Area Representing a c Confidence Interval on  2 Distribution with d.f. = n - 1 Area = c

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 A c Confidence Interval for  2

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 A c Confidence Interval for 

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 A sample of twenty-four adults has a standard deviation of s = 16.5 for serum cholesterol. Find a 95% confidence interval for the population variance,  2.

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 A sample of twenty-four adults has a standard deviation of s = 16.5 for serum cholesterol. c = 0.95 n = 24 d.f.= n - 1 = 23 s = 16.5

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 A c Confidence Interval for  2

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 95% Confidence Interval for Variance We are 95% confident that the true population variance falls between 164.44 and 535.65

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 95% Confidence Interval for Standard Deviation To find a confidence interval for standard deviation, take square roots. We are 95% confident that the population standard deviation falls between 12.82 and 23.14.