1. Chapter 11 Chi-Square Distribution

2. Review So far, we have used several probability distributions for hypothesis testing and confidence intervals with normal distribution and Student’s t distribution. In this section, we will be using chi-squre.

3. What is Chi-Square?

4. Example:

5. Mode (high point)

7. Degrees of Freedom

8. Example: (The situation) Innovative Machines Incorporated has developed two new letter arrangements for computer keyboards. The company wishes to see if there is any relationship between the arrangement of letters on the keyboard and the number of hours it takes a new typing student to learn to type at 20 words per minute. Or, from another point of view, is the time it takes a student to learn to type independent of the arrangement of the letters on a keyboard? Use 5% level of significance

9. Example: (step 1)

10. Example: (chart) Step 2: Determine E

11. Answer for E (will show in class) Keyboard21-40 h41-60 h61-80 hRow Total AO:25 E:24 O:30 E:40 O:25 E:16 80 BO:30 E:36 O:71 E:60 O:19 E:24 120 StandardO:35 E:30 O:49 E:50 O:16 E:20 100 Column Total9015060300 (sample size)

12. Cell 12524110.04 23040-101002.50 325169815.06 43036-6361.00 57160111212.02 61924-5251.04 735305250.83 8495010.02 91620-4160.80

13. 0.04 2.50 5.06 1.00 2.02 1.04 0.83 0.02 0.80

14. Example: (Degrees of freedom for test of independence)

15. Conclusion Look in the book with chi-square table. Since we have Chi-square as 13.31 with d.f. 4 The corresponding P-value falls between 0.005 and 0.010. Since (.005< P-Value < 0.010) <.05, we reject null and accept alternate. Based on 5% level of significance, we are taking a chance to conclude that keyboard arrangement and learning time are not independent.

16. Group Work (the situation) Vending Machine is to install soda machines in elementary school and high school. The market analyst wish to know if flavor preference and school level are independent. A random sample of 200 students was taken. Their school level and soda preferences are given. Is independence indicated at the 1% level of significance?

17. Group Work (table) SodaHigh SchoolElementaryRow Total CokeO:33 E: O:57 E: 90 PepsiO:30 E: O:20 E: 50 Mountain DewO:5 E: O:35 E: 40 FantaO:12 E: O:8 E: 20 Column Total80120200 (sample size)

18. How to Test for independence of two statistical variables Look at Pg 582. Copy it and follow it!

19. Test of homogeneity The test claim that different populations share the sample proportions of specified characteristics.

20. Test of Homogeneity

21. Example: If you could own one pet, what kind would you choose? The possible responses were of the following. Does the same proportion of males same as females prefer each type of pet? Use 1 % level of significance GenderDogCatOther petNo Pet Female1201321830 Male135702025

22. Fill this out GenderDogCatOther petNo PetRow Total FemaleO:120 E: O:132 E: O:18 E: O:30 E: MaleO:135 E: O:70 E: O:20 E: O:25 E: Column Total

23. Answer GenderDogCatOther petNo PetRow Total FemaleO:120 E:139.09 O:132 E:110.18 O:18 E:20.73 O:30 E:30 300 MaleO:135 E:115.91 O:70 E:91.82 O:20 E:17.27 O:25 E:25 250 Column Total 2552023855550 (sample size)

24. Fill this out Cell 1 2 3 4 5 6 7 8

25. Answer Cell 1120139.092.62 2132110.184.320 31820.730.359 430 0 5135115.913.144 67091.825.185 72017.270.431 825 0

26. Final Answer Chi-square= 16.059 d.f.=3 P-value=.001 Based on 1% level of significance, we are taking a chance to say that males and female students have different preferences when it comes to selecting a pet because we rejected the null saying preference is the same and accept the alternate saying the preference is different.

27. Homework Practice Pg 588 #1-15 even

28. CHI-SQUARE: GOODNESS OF FIT

29. Reason Behind Goodness of Fit Set up a test to investigate how well a sample distribution fits a given distribution Use observed and expected frequencies to compute the sample chi-square statistics Find or estimate the P-value and complete the test

30. Hypothesis Testing

31. Sample statistic

32. Question Does present distribution of favorable responses the same or different than last year? To test this hypothesis, a random sample of 500 employees was taken. The chart is on the next slide. Use 1% level of significance

33. Example CategoryPercentage of Favorable Responses Vacation time4% Salary65% Safety regulations13% Health and retirement benefits12% Overtime policy and pay6% CategoryObserved Vacation time30 Salary290 Safety regulations70 Health and retirement benefits 70 Overtime40

34. Answer CategoryOE Vacation time30201005.00 Salary29032512253.77 Safety regulations 7065250.38 Health and retirement benefits 70601001.67 Overtime40301003.33 Total500 14.15

35. Answer

36. Group Work The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community (Red Lake village) Use 5% level of significance to test the claim that the age distribution fits the age distribution of red lake village Age% populationObserved in Red Lake Village Under 57.2%47 5-1413.6%75 15-6467.1%288 65 +12.1%45

37. Answer

38. Homework Practice Pg 597 #1-18 even