1. Statistical Inferences for Variance Objectives: Learn to compare variance of a sample with variance of a population Learn to compare variance of a sample with variance of another sample Introduce chi-squared distribution Introduce F-distribution

2. If a manager were trying to make a process output less variable, they could take a sample from the changed process and compare it with known data from the old process. It can then be determined if the new process is significantly less variable than the old process. Comparing a Sample Variance with a Population Variance

3. Null hypothesis is that the new variance is equal to the old variance Alternative hypothesis is that the new population variance is smaller than the old variance One sided test Comparing a Sample Variance with a Population Variance

4. Chi-squared distribution used to compare a sample variance with a population variance: Comparing a Sample Variance with a Population Variance

5. Comparing Two Sample Variances If there are two samples with corresponding variances, is one sample variance significantly different from the other? Or, is it reasonable to say that both sample variances came from the same population?

6. Comparing Two Sample Variances The appropriate test is the F-test, or Variance- ratio test. This test uses the ratio of the two sample variances: where s 1 is the larger estimate of variance.

7. Comparing Two Sample Variances Null Hypothesis: the difference between the two estimates of variance is entirely due to chance, so. Alternative Hypothesis:. Because the larger variance is placed in the numerator, there is no reason to consider:.

8. The F-test determines whether one set of data is significantly more variable than another data set. This determination can be used for two main purposes: 1. A two tailed F-test can indicate whether or not two estimates of variances are similar enough at the 10% level of significance to be pooled, for comparing means using an unpaired t-test. Comparing Two Sample Variances

9. 2. A one tailed F-test can be used to help determine whether the variation of treatment or conditions has an effect on the results. If the data from an experiment can be broken down into groups, where each group is made up of data obtained using the same conditions and repeated measurements. Then the variance within groups can be compared with the variance between groups. The variance between groups is caused by different treatments or conditions and experimental error. Comparing Two Sample Variances