1. Section 6.4 Inferences for Variances

2. Chi-square probability densities

3. Use of Chi-square distribution Sample variance S 2 is used to estimate the population variance  2 Assume that the population is normal. has a chi-squared distribution with v=n-1 degrees of freedom. Chi-square probabilities can be found in table B.5.

4. Then we get a 95% confidence interval: Confidence interval for  2

5. Example The soft-drink company wants to control the variability in the amount of fill. A sample of size 28 was drawn and the sample variance s 2 =0.0007. Give a 95% confidence interval about the variance  2.

6. Solution v=n-1=27. What is the CI for  ?

8. Example The soft-drink company wants to control the variability in the amount of fill. A sample of size 28 was drawn and the sample variance s 2 =0.0007. Use the five-step significance testing format to assess the evidence that the variance is greater than 0.0005.

9. 1 &2. 3. The test statistic is 4. The sample gives 5. The observed level of significance is P(a chi- square random variable with 27 d.f>37.8) 0.05<P-value<0.1

10. Inference for the Ratio of two variances For two independent samples from normal distributions, sometimes we want to compare two variances. A new distribution called F distribution can be of use here.

11. F distribution

12. F-distribution A F-distribution has two degrees of freedom: – Numerator degrees of freedom v1 – Denominator degrees of freedom v2 – Tables B.6 give quantiles of F-distributions

13. For F-distribution So This is particularly useful since in table B.6 only quantiles for p larger than 0.5 are given.

14. The ratio of two variances When s 1 2 and s 2 2 come from independent samples from normal distributions, the variable where n 1 -1 and n 2 -1 are associated degrees of freedom for s 1 and s 2.

15. Confidence interval for   2   2

20. A five step significance test of equality of variances: