STA 6166 UNIT 2 Section 3
|Welcome||<||Begin||<||Unit 2||<||Section 3||>||Section 3 Exercises|
|Readings||Ott and Longnecker, Chapter 7, pages 341-378.|
In the previous two sections we studied statistical tests for the population mean for one independent population and methods for comparing the population mean for two independent populations. At some point you probably wondered, "What about population variances?" Especially in the case of two populations, it would be nice to have a statistical test to decide whether you should be using a pooled variance t-test or a separate variance t-test.
Because the sample variance is a sum of squared values, the Central Limit Theorem for the Mean does not hold hence we cannot depend on it to specify the sampling distribution for the variance. It turns out that the sampling distribution for sums of squares follows a Chi Square distribution. This is a non-symmetric distribution that is only defined from positive real numbers and has one parameter we refer to as the degrees of freedom. With the Chi Square distribution we can construct confidence intervals for the variance and it forms the basis for testing whether the true population variance is equal to some specified value in the one population case.
Testing for equality of the variances of two independent populations is not as straightforward as that for means. Instead of taking the difference between the variances, the test statistic is the ratio of the variances. The use of this test statistic follows from the fact that we can determine the distribution of the ratio of Chi Square distributed random variables, but we cannot easily determine the distribution of the difference of two Chi Square distributed random variables. The ratio follows an F-distribution. This is another non-symmetric distribution defined only for positive real numbers. The F-distribution is indexed by two parameters, typically referred to as the numerator degrees of freedom and denominator degrees of freedom.
The Chi Square test for a variance and the F-test for the ratio of two variances are easily to perform. The most difficult part of the test is determining the critical values from the appropriate tables in the Appendix (pages1100-1114). These tables are not like the normal and t-distribution tables and take some practice to use.
There are a number of test for comparing variances across more than two populations. Hartley's Fmax Test for Homogeniety of Population Variances is probably the simplest of these tests to perform. Unfortunately it is not necessarily the most powerful. Levene's test and the other tests presented in the lecture notes are more powerful.
Comparing variances is most often performed to determine if the condition of homogeniety of variances (equal variances) holds for the populations being examined. This will be important for checking this assumption for the simple Analysis of Variance techniques in the next Unit.
|PPT Lecture||Inferences about Variances (PowerPoint, PDF)|
|Exercises||To check your understanding of the readings and practice these concepts and methods, go to Unit 2 Section 3 Exercises, do the exercises then check your answers from the page provided. Following this continue on to the Unit 2 Test .|