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Tests for Variances. (Session 11). 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
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Tests for Variances (Session 11)
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
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
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?
Why study variance? • 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
Testing for a single variance • 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
Comparing variances • 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?
Test for comparing variances • 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.
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?
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
Conducting the test • Calculate the chi-square test statistic Under H0, 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.
Form of Chi-square distribution Shaded area represent the p-value Value of calculated test-statistic
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?
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.
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
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).
Example of an F-distribution 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. 1% region (0.5% x2) 0.49 2.11
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?
Computations • The value of the F-statistic is • The p-value is 0.01 (from statistics software). This indicate a significant difference in variance.
Tests for comparing several variances • Levene’s 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 • Bartlett’s test – based on a chi-square statistic. The test is dependent on meeting the assumption of normality. It is therefore weaker than Levene’s test.
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.)
