Chapter 15 The Analysis of Qualitative Data

1. DoingStatistics for BusinessData, Inference, and Decision MakingMarilyn K. PelosiTheresa M. Sandifer Chapter 15 The Analysis of Qualitative Data

2. DoingStatistics for Business Chapter 15 Objectives In this chapter you will learn to use chi- square tests for: • Testing whether a particular probability model fits a set of data (Goodness of Fit test) • Testing equality of proportions for more than 2 populations

3. DoingStatistics for Business Chapter 15 Objectives (con’t) • Testing whether 2 qualitative variables are dependent or independent

4. The chi-square goodness of fit test checks to see how well a set of data fit the model for a particular probability distribution. Doing Statistics for Business

5. The observed frequencies are the actual number of observations that fall into each class in a frequency distribution or histogram. Doing Statistics for Business

6. The expected frequencies are the number of observations that should fall into each class in a frequency distribution under the hypothesized probability distribution. Doing Statistics for Business

7. A uniform distribution is one in which each outcome or class of outcomes is equally likely to occur. Doing Statistics for Business

8. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Setting Up the Hypotheses for the Goodness of Fit Test Analysts for insurance companies assume that the number of drivers who wear seat belts is a binomial random variable with  = 0.70. To test this assumption they decide to set up checkpoints and sample 10 drivers every 2 hours. Set up the hypotheses to perform an appropriate chi-square goodness of fit test.

9. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Calculating the Expected Frequencies The insurance analysts collect data for 1000 samples of 10 drivers and obtain the frequency distribution shown in the following table. Find the expected frequency distribution for the data if the distribution is really binomial with n = 10 and  = 0.70.

10. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Calculating the Expected Frequencies (con’t)

11. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Calculating the Expected Frequencies (con’t) Create frequency histograms for both the observed and the expected frequency distributions. At this point, does it appear that the observed data conform to the binomial distribution with n = 10 and  = 0.70.? Why or why not?

12. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Calculating the Chi-Square Statistic Fill in the following table to calculate the value of the chi-square statistic for the data obtained by the insurance analysts.

13. DoingStatistics for Business Figure 15.1 The Upper Tail of a Chi-Square Distribution

14. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Finding the Critical Value and Performing the Test The insurance analysts decide that they want to test the goodness of fit hypothesis at the 0.01 level of significance. How many degrees of freedom will the critical value for the test have? Find the critical value for the test.

15. DoingStatistics for Business TRY IT NOW! Seat-Belt Usage Finding the Critical Value and Performing the Test (con’t) Based on the chi-square test statistic and the critical value, what can you conclude about the distribution of the number of people in a sample size of 10 that wear seat belts?

16. DoingStatistics for Business TRY IT NOW! Technical Support Setting Up the Chi-Square Test for Proportions A company that sells computer software has three different locations set up to provide customers with technical support for their products. The support representatives keep a log for each call to technical support, and as part of that log, they record whether the problem was resolved successfully.

17. DoingStatistics for Business TRY IT NOW! Technical Support Setting Up the Chi-Square Test for Proportions (con’t) The company analysts are interested in knowing whether the percentage of calls that are successfully resolved is the same for each location. They randomly select logs from each location and collect data on the number of calls that result in a successful resolution of the problem. The data are summarized in the following table.

18. DoingStatistics for Business TRY IT NOW! Technical Support Setting Up the Chi-Square Test for Proportions (con’t) Set up the hypotheses for the software company. Calculate the proportion of successfully resolved calls for each location. Based solely on these numbers, do you think that the proportion of successfully resolved calls for all three locations is the same?

19. DoingStatistics for Business TRY IT NOW! Technical Support Calculating the Expected Frequencies The data for the computer software company interested in its technical support locations are

20. DoingStatistics for Business TRY IT NOW! Technical Support Calculating the Expected Frequencies Calculate , the percentage of calls that are resolved successfully, assuming that the three locations are the same. Use the overall proportion of successful calls to find the expected frequency of successful calls for each location.

21. DoingStatistics for Business Figure 15.2 Minitab Output from Chi-Square Analysis

22. DoingStatistics for Business TRY IT NOW! Technical Support Performing the Chi-Square Test for Proportions The computer software company with the different technical support location wants to complete the test to determine whether the percentage of successfully resolved calls is the same at all three locations. It wants to test at the 0.01 level of significance.

23. DoingStatistics for Business TRY IT NOW! Technical Support Performing the Chi-Square Test for Proportions (con’t) Calculate the value of the chi-square test statistic and complete the test. Is the proportion of successfully resolved calls the same at each location?

24. DoingStatistics for Business TRY IT NOW! Drinking Survey Setting Up the Contingency Table for a Test for Independence The Public Health student who did the study on drinking also collected data on the number of times that the student drove while intoxicated in the last two weeks (coded). The contingency table for the usable responses is given on the following slide.

25. DoingStatistics for Business TRY IT NOW! Drinking Survey Setting Up the Contingency Table for a Test for Independence (con’t)

26. DoingStatistics for Business TRY IT NOW! Drinking Survey Setting Up the Contingency Table for a Test for Independence (con’t) The university is interested in knowing if the number of times a student drove while intoxicated is related to his or her class in school. It feels that this information will help target student audiences for programs on drinking and driving.

27. DoingStatistics for Business TRY IT NOW! Drinking Survey Setting Up the Contingency Table for a Test for Independence (con’t) Set up the hypotheses that the university should test. Calculate the expected frequencies for each cell and put them in the appropriate location in the table. Are any of the expected frequencies less than 5? If so, can you suggest a logical way to combine categories to avoid this problem?

28. DoingStatistics for Business TRY IT NOW! Drinking Survey Performing the Chi-Square Test for Independence The university that is looking at the relationship between class year and drinking and driving wants to perform the test at the 0.05 level of significance. Calculate the value of the chi-square statistic for the data. Find the critical value and perform the test. Are class and drinking and driving independent?

29. Doing Statistics for Business The Chi-Square Test for Independence in Excel - Creating the Contingency Table 1. From the Data menu, select Pivot Table Report and follow the steps of the pivot table wizard to create the table. 2. At step 4 of 4, after you indicate where you want the table to be placed, do not select Finish. Instead, clickOptions. 3. Make sure that the checkbox next to For empty cells, show is checked and in the textbox next to it, type “0”. (Excel will not accept empty cells in the contingency table.) 4. Click OK and then Finish.

30. Doing Statistics for Business Figure 15.4 Pivot Table Options Dialog Box

31. Doing Statistics for Business Figure 15.5 Contingency Table for Binge Drinking Data

32. Doing Statistics for Business The Chi-Square Test for Independence In Kadd Select Hypothesis Testing>Chi Square Test from the Kadd menu The dialog box opens

33. Doing Statistics for Business The Chi-Square Test for Independence Dialog Box

34. Doing Statistics for Business The Chi-Square Test for Independence Complete the Dialog Box: • The input range should be the location of the contingency table for the data. Make sure you do not select the column and row that contain the totals. • Indicate where you want the output located. • Click OK.

35. Doing Statistics for Business The Chi-Square Test for Independence

36. Doing Statistics for Business Chapter 15 Summary In this chapter you have learned: • The chi-square test involves comparing observed and expected frequencies for different classes of data. • The chi-square test is quite versatile and can be used to test:

37. Doing Statistics for Business Chapter 15 Summary (con’t) Goodness of Fit Equality of Proportions for more than 2 Populations Independence of Qualitative Variables • The results of a chi-square test do not solve a problems, but simply point out when further action is indicated and when it is not.