1. SADC Course in Statistics Tests for Variances (Session 11)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session, you will be able to: identify situations where testing for a population variance, or comparing variances, may be applicable conduct a chi-square test for a single population variance or an F-test for comparing two population variances interpret results of tests concerning population variances

3. To put your footer here go to View > Header and Footer 3 Session Contents In this session you will be shown why in some applications the study of population means alone is not adequate be introduced to the chi-square and F tests for testing one population variance and comparing two population variances respectively see examples of the applicability of these tests

4. To put your footer here go to View > Header and Footer 4 Is the packing machine working properly? Suppose people have lodged complaints about the weight of the 12.5 Kg mealie- meal bags. A consultant took a sample of mealie-meal bags and did not find any problem with the average weight. That is, she could not reject the null hypothesis that the population mean weight = 12.5 Kg What could be the problem?

5. To put your footer here go to View > Header and Footer 5 Although the mean is OK in the above example, there could be a problem with the variance Packaging plants are designed to operate within certain specified precision Ideally it would be desirable to have the machine pack exactly 12.5 Kg in every bag but this is practically impossible. So a certain pre-specified variation is tolerated Why study variance?

6. To put your footer here go to View > Header and Footer 6 After years of operation it is always important to check whether the machine variation is still at the initially set level of precision (say ) This implies testing the hypothesis against the alternative Testing for a single variance

7. To put your footer here go to View > Header and Footer 7 A similar problem could occur if a factory manager is considering whether to buy packaging Machine A or Machine B. During test runs, Machine A produced sample variance while Machine B produced sample variance. Question: Are these variances significantly different? Comparing variances

8. To put your footer here go to View > Header and Footer 8 Suppose the population variances for weights of mealie-meal bags packaged from machines A and B are respectively and. We can answer the question concerning whether the variances are different by testing the null hypothesis against the alternative. We will return to this later in the session. Test for comparing variances

9. To put your footer here go to View > Header and Footer 9 Other applications Other applications where testing for variance may be important includes the following: Foreign exchange stability is important in any economy. Too much variation of a currency is not good. Price stability of other commodities is also important. Question: Can you name other possible areas of application where testing that the variation remains stable at a pre-set value is important?

10. To put your footer here go to View > Header and Footer 10 The chi-square test This test applies when we want to test for a single variance. The null hypothesis is of the form Need to test this against the alternative The test is based on the comparison between and using the ratio

11. To put your footer here go to View > Header and Footer 11 Calculate the chi-square test statistic Under H 0, this is known to have a chi- square distribution with n-1 degrees of freedom. Compare this with chi-square tables, or use statistics software to get the p-value. Here, p-value = where is a chi-square random variable. Conducting the test

12. To put your footer here go to View > Header and Footer 12 Form of Chi-square distribution Value of calculated test-statistic Shaded area represent the p-value

13. To put your footer here go to View > Header and Footer 13 Back to Example Suppose the mealie-meal packaging machine is designed to operate with precision of. Suppose that data from a sample of 12 mealie-meal bags gave. Does the data indicate a significant increase in the variation?

14. To put your footer here go to View > Header and Footer 14 Test computations and results The calculated chi-square value The p-value (based on a chi-square with 11 d.f.) is indicating no significant increase in the variance.

15. To put your footer here go to View > Header and Footer 15 The F-test The F-test is used for comparing two variances, say and. The hypothesis being tested is with either a one-sided alternative ; ; or a two-sided alternative

16. To put your footer here go to View > Header and Footer 16 The F-test The null hypothesis is rejected, for large values of the F-statistic below, in the case of a one-sided test For a 2-sided test, need to pay attention to both sides of the F-distribution (see below).

17. To put your footer here go to View > Header and Footer 17 Example of an F-distribution 1% region (0.5% x2) 0.49 2.11 To use just the upper tail value, ensure F-ratio is calculated so it is >1, then use upper tail of the 2½% F-tabled value when testing at 5% significance.

18. To put your footer here go to View > Header and Footer 18 Numerical Example Suppose 20 items produced on test trial of Machine A gave while 27 items produced by Machine B gave Does the data provide evidence that the working precision of the two machines are significantly different?

19. To put your footer here go to View > Header and Footer 19 Computations The value of the F-statistic is The p-value is 0.01 (from statistics software). This indicate a significant difference in variance.

20. To put your footer here go to View > Header and Footer 20 Tests for comparing several variances Levenes test – is robust in the face of departures from normality. It is used automatically in some software before conducting other tests which are based on the assumption of equal variance Bartletts test – based on a chi-square statistic. The test is dependent on meeting the assumption of normality. It is therefore weaker than Levenes test.

21. To put your footer here go to View > Header and Footer 21 References Gallagher, J. (2006) The F-test for comparing two normal variances: correct and incorrect calculation of the two-sided p-value. Teaching Statistics, 28, 58-60. (this gives an example to show that some statistics software packages can give incorrect p-values for F-values close to 1.)

22. To put your footer here go to View > Header and Footer 22 Some practical work follows…