Statistical Tests: Chi-Square Application & Interpretation

1. CHAPTER 11 • Chi- square goodness of fit Test “GOF” • Chi-square Test for Homogeneity • Chi-square Test for Independence • Symbol for Chi-Square is χ2 DONE SAME WAY

2. What’s the difference? • Goodness of Fit…..how well does the observed data match the expected….2 rows or 2 columns • Homogeneity…..More than one sample is taken with one categorical variable in mind • (2+ Samples, 1 category) • Independence/Association…..Only one sample is taken and there are two or more categories. • (1 sample, 2+ categories)

3. Chi-Square Curve It is not a NORMAL CURVE!!! It is always skewed to the right some

4. Is your die fair—one more time. Roll your die 60 times. Write down the number for every roll.

5. Chi-Square GOF Test • If your die is fair you would expect to get 10 of each number in 60 rolls. • In this test we compare the EXPECTED results vs the OBSERVED results.

6. Hypotheses • Ho: The proportion of each number that occurs on my die is 1/6 • Ha: The proportion of each number that occurs on my die is different than 1/6 • There are no symbols for the chi-square. However, it is always one-sided, even though the word “different” is used.

7. Normal condition for χ2 • 80% of the expected cells are greater than or equal to 5. (not observed cells—expected cells!)

8. Formula

9. Degrees of Freedom (df) • For all chi-square tests use the following: • df = (r – 1)(c – 1) • r is the row and c is the column

10. Ho: The distribution of my die results are as EXPECTED Ha: The distribution of my die results are different than EXPECTED Assumptions: We have an independent random sample of 60 rolls of my fake die. 100% of our expected cells are 5 or more. Chi-square GOF test This p-value is ___________to reject Ho at _______ level There is_____________evidence to suggest that my die results are ___________ than expected. X2 = P-value = df =

11. Ho: Each groups fastest time occurred on each lane as expected Ha: Each groups fastest time occurred on each lane differently than EXPECTED Assumptions: We have an independent random sample of 21 groups fastest time and the lane it occurred on. We can assume that there are at least 210 cars of this type in the population. 100% of our expected counts are 5 or more. Chi-square GOF test This p-value is ___________to reject Ho at _______ level There is_____________evidence to suggest that the lanes are different than expected. X2 = P-value = df =

12. Calculator steps TI-83+ Calculator Then hit 2nd and Stat Put your observed counts in L1 and Expected in L2 Tyoe the observed in L1 and expected in L2. Then click on the L3 heading and type the formula(then click enter), then quit out to the main screen Find the sum of L3, your answer is the chi-square statistic

13. Calculator steps After your get the sum you need to obtain the p-value X2 , UB , df This will give you the p-value

14. Calculator steps TI-84 calculator does most of the work for you Make sure you have typed your observed counts in L1 and expected counts in L2 5

17. Demographics. • Rancho is approximately 54.4% Hispanic, 29.7% Asian, 11.4% white and 4.5% other.(data as of 2013-2014 school year) • Does Mr. Pines‘AP stats classes reflect this diversity? Run the appropriate test, verify your requirements, and write a conclusion.

18. Demographics. 2016 Ho: The diversity in Mr. Pines classes is the same as Rancho’s diversity Ha: The diversity in Mr. Pines classes is different than Rancho’s diversity Assumptions: We have an independent random sample of ___ students ethnicity. We can assume that there have been at least _____ students in Mr. Pines classes. 100% of our expected cells are 5 or more. Chi-square GOF test This p-value is_________to reject Ho at the _____ level This is _______evidence to suggest that Mr. Pines class diversity may be different than Rancho’s diversity. X2 = P-value =. df =

19. Demographics. 2015 Ho: The diversity in Mr. Pines classes is the same as Rancho’s diversity Ha: The diversity in Mr. Pines classes is different than Rancho’s diversity Assumptions: We have an independent random sample of 159 students ethnicity. We can assume that there have been at least 1590 students in Mr. Pines classes. 100% of our expected cells are 5 or more. Chi-square GOF test This p-value is low enough to reject Ho at the 1% level This is strong evidence to suggest that Mr. Pines class diversity may be different than Rancho’s diversity. X2 = 12.38 P-value =.0062 df = 3

20. Is there a difference…… • Do boys and girls prefer types of social media? • Please choose your favorite of the three below.

21. One Hundred Twenty Nine students were surveyed…………. We took a sample of 72 girls and a sample of 57 boys. c/o 2016

22. One Hundred Fifty Seven students were surveyed…………. We took a sample of 83 girls and a sample of 74 boys. c/o 2015

23. What’s the difference? • Goodness of Fit…..how well does the observed data match the expected….2 rows or 2 columns • Homogeneity…..More than one sample is taken with one categorical variable in mind • (2+ Samples, 1 category) • Independence/Association…..Only one sample is taken and there are two or more categories. • (1 sample, 2+ categories)

24. Hypotheses for Chi-Square Test for Homogeneity Ho: There is no difference between gender and social media preference Ha: There is a difference between gender and social media preference OR Ho: The proportions of boys and girls who prefer each type of social media are the same Ha: The proportions of boys and girls who prefer each type of social media are different

25. Ho: There is no difference between gender and social media preference Ha: There is a difference between gender and social media preference Assumptions: We have two independent random samples of students social media preferences(72 girls and 57 boys). There are obviously more than 720 girls and 570 boys in the population sampled from who use social media. 100% of our expected cells are 5 or more. Chi-square test of homogeneity This p-value is _______to reject Ho at _______ level There is___________evidence to suggest that there may be a difference between gender and social media preference. X2 = P-value =. df =

26. Period 1

27. Period 3

28. Referrals vs Days of week The table shows the number of students referred for disciplinary reasons to the principals office, broken down by day of the week. Are referrals related to the day of the week?

30. BIRTHDAYS. • Are Mr. Pines students birth months distributed in proportion to the number of days in each month? • We can run a chi-square GOF based on the # of days in each month n = 153

31. C/O 2015 per 1 only H0: Mr. Pines students birthday months are in proportion to the number of days in each month. Mr. Pines students birthday months are different than the proportion of the number of days in each month. Ha: We have an independent sample of 53 students birth months. All births is obviously more than 10x our sample. 100% of our expected counts are 5 or more. Chi-square GOF Test X2 = 9.81 P-value = .5475 df = 11 This p-value is too high to reject Ho Based on this sample, there is NOT enough evidence to suggest that Mr. Pines students birthday months are different than the proportion of the number of days in each month.

32. BIRTHDAYS To figure out the expected we need to think about the number of days in each month. Jan 31 July 31 Feb 28 Aug 31 Mar 31 Sep 30 Total = 365 days Apr 30 Oct 31 May 31 Nov 30 June 30 Dec 31 n = 153

33. C/O 2016 H0: Births are distributed according to the # of days in each month Ha: Births are not distributed according to the # of days in each month Assumptions: We have an independent sample of ____ students birth months. All births is obviously more than 10x our sample. 100% of our expected counts are 5 or more. X2 = P-value =. df = Chi-square GOF test This p-value is ________ to reject Ho at _______level. There is ___________evidence to suggest that births are not uniformly distributed by the # of days in each month.

34. C/O 2014 H0: Births are uniformly distributed by the # of days in each month Ha: Births are not uniformly distributed by the # of days in each month CONDITIONS We have an independent sample of 129 students birth months. All births is obviously more than 10x our sample. 100% of our expected counts are 5 or more. df = 11 X2 = 7.75 P = .7355 This p-value is too high to reject Ho. There is not enough evidence to suggest that births are not uniformly distributed by the # of days in each month.

35. Chi-Square Test for Homogeneity • Data is given in a 2-way table • Expected counts are found by using a matrix on your calculator or by multiplying the (ROW TOTAL)(COLUMN TOTAL)/GRAND TOTAL • Conditions and df are the same as GOF test

36. Is there a difference…… • Do boys and girls prefer different video game consoles? • Please choose your favorite console out of the 3?

37. Hypotheses for Chi-Square Test for Homogeneity Remember Ha is always means different! Ho: There is no difference between gender and video game console preference Ha: There is a difference between gender and video game preference OR Ho: The proportions of boys and girls who prefer each type of console are the same Ha: The proportions of boys and girls who prefer each type of console are different

38. Why is this a Homogeneity Test? • Two samples were taken separately • Boys console preference • Girls console preference • There is ONE category of interest.

39. Two-Way Table 2013

40. Two-Way Table 2014

41. Quit to home screen, go to test menu Hit 2nd Matrix, go to EDIT Should be already setup if you used A and B Set the appropriate matrix size Enter observed counts in matrix

42. You need the expected counts….so go back to 2nd matrix. Use NAMES and go down to Matrix B, calculator generates them after you run the test You will most likely have to scroll to the right to see all of the expected counts REMEMBER!....EXPECTED COUNTS MUST BE ON YOUR PAPER!

44. What’s the difference? • Goodness of Fit…..how well does the observed data match the expected….2 rows or 2 columns • Homogeneity…..More than one sample is taken with one categorical variable in mind • (2+ Samples, 1 category) • Independence/Association…..Only one sample is taken and there are two or more categories. • (1 sample, 2+ categories)

45. College Students’ Drinking In 1987, a random sample of undergraduate students at Rutgers University was sent a questionnaire that asked about their alcohol drinking habits. Here are the results displayed in a two-way table.

46. Chi-Square Test for Independence There was one sample taken and then data was broken down into different categories. When only one sample is taken we are doing a Chi-Square Test for Independence/Association

47. Hypotheses This is a chi-square test for Independence/Association, you have a few options for writing the hypotheses Ho: There is no association between students’ residence type and drinking habits. Ha There is an association between students’ residence type and drinking habits. OR Ho: Student drinking habits and residence type are independent Ha Student drinking habits and residence type are not independent

48. Full Moon Some people believe that a full moon elicits unusual behavior in people. The table shows the number of arrests made in a small town during the weeks of six full moons and six other randomly selected weeks in the same year. Is there evidence of a difference in the types of illegal activity that takes place. This is a chi-square test for Homogeneity

49. Thanks To: Grace Montgomery

50. THANKS TO: Amy Nguyen