1. Copyright © Cengage Learning. All rights reserved. 9 Inferences Based on Two Samples

2. Copyright © Cengage Learning. All rights reserved. 9.5 Inferences Concerning Two Population Variances

3. 3 Methods for comparing two population variances (or standard deviations) are occasionally needed, though such problems arise much less frequently than those involving means or proportions. For the case in which the populations under investigation are normal, the procedures are based on a new family of probability distributions.

4. 4 The F Distribution

5. 5 The F probability distribution has two parameters, denoted by v 1 and v 2. The parameter v 1 is called the number of numerator degrees of freedom, and v 2 is the number of denominator degrees of freedom; here v 1 and v 2 are positive integers. A random variable that has an F distribution cannot assume a negative value. Since the density function is complicated and will not be used explicitly, we omit the formula. There is an important connection between an F variable and chi-squared variables.

6. 6 The F Distribution If X 1 and X 2 are independent chi-squared rv’s with v 1 and v 2 df, respectively, then the rv (the ratio of the two chi-squared variables divided by their respective degrees of freedom), can be shown to have an F distribution. (9.8)

7. 7 The F Distribution Figure 9.7 illustrates the graph of a typical F density function. Figure 9.7 An F density curve and critical value

8. 8 The F Distribution Analogous to the notation t ,v and we use for the value on the horizontal axis that captures  of the area under the F density curve with v 1 and v 2 df in the upper tail. The density curve is not symmetric, so it would seem that both upper- and lower-tail critical values must be tabulated. This is not necessary, though, because of the fact that

9. 9 The F Distribution Appendix Table A.9 gives for  =.10,.05,.01, and.001, and various values of v 1 (in different columns of the table) and v 2 (in different groups of rows of the table). For example, F.05,6,10 = 3.22 and F.05,10,6 = 4.06. The critical value F.95,6,10, which captures.95 of the area to its right (and thus.05 to the left) under the F curve with v 1 = 6 and v 2 = 10, is F.95,6,10 = 1/F.05,10,6 = 1/4.06 =.246.

10. 10 The F Test for Equality of Variances

11. 11 The F Test for Equality of Variances A test procedure for hypotheses concerning the ratio is based on the following result. Theorem

12. 12 The F Test for Equality of Variances This theorem results from combining (9.8) with the fact that the variables and each have a chi-squared distribution with m – 1 and n – 1 df, respectively. Because F involves a ratio rather than a difference, the test statistic is the ratio of sample variances. The claim that is then rejected if the ratio differs by too much from 1.

13. 13 The F Test for Equality of Variances Recall that the P-value for an upper-tailed t test is the area under an appropriate t curve to the right of the calculated t, whereas for a lower-tailed test the P-value is the area under the curve to the left of t. Analogously, the P-value for an upper-tailed F test is the area under an appropriate F curve (the one with specified numerator and denominator dfs) to the right of f, and the P- value for a lower-tailed test is the area under an F curve to the left of f.

14. 14 The F Test for Equality of Variances Because t curves are symmetric, the P-value for a two- tailed test is double the captured lower tail area if t is negative and double the captured upper tail area if t is positive. Although F curves are not symmetric, by analogy the P- value for a two-tailed F test is twice the captured lower tail area if f is below the median and twice the captured upper tail area if it is above the median.

15. 15 The F Test for Equality of Variances

16. 16 The F Test for Equality of Variances

17. 17 The F Test for Equality of Variances

18. 18 The F Test for Equality of Variances

19. 19 The F Test for Equality of Variances

20. 20 The F Test for Equality of Variances

21. 21 The F Test for Equality of Variances The F tests discussed in succeeding chapters will all be upper-tailed. If, however, a lower-tailed F test is appropriate, then lower- tailed critical values should be obtained as described earlier so that a bound or bounds on the P-value can be established. In the case of a two-tailed test, the bound or bounds from a one-tailed test should be multiplied by 2.

22. 22 The F Test for Equality of Variances

23. 23 The F Test for Equality of Variances Various statistical software packages will, of course, provide an exact P-value for any F test.

24. 24 Example 9.14 A random sample of 200 vehicles traveling on gravel roads in a county with a posted speed limit of 35 mph on such roads resulted in a sample mean speed of 37.5 mph and a sample standard deviation of 8.6 mph, whereas another random sample of 200 vehicles in a county with a posted speed limit of 55 mph resulted in a sample mean and sample standard deviation of 35.8 mph and 9.2 mph, respectively (these means and standard deviations were reported in the article “Evaluation of Criteria for Setting Speed Limits on Gravel Roads” (J. of Transp. Engr., 2011: 57–63); the actual sample sizes result in dfs that exceed the largest of those in our F table).

25. 25 Example 9.14

26. 26 Example 9.14

27. 27 Example 9.14 7. The P-value clearly exceeds the mandated significance level. The null hypothesis therefore cannot be rejected; it is plausible that the two speed distribution variances are identical. The sample sizes in the cited article were 2665 and 1868, respectively, and the P-value reported there was.0008. So for the actual data, the hypothesis of equal variances would be rejected not only at significance level.10—in contrast to our conclusion—but also at level.05,.01, and even.001.

28. 28 Example 9.14 This illustrates again how quite large sample sizes can magnify a small difference in estimated values. Note also that the sample mean speed for the county with the lower posted speed limit was higher than for the county with the lower limit, a counterintuitive result that surprised the investigators; and because of the very large sample sizes, this difference in means is highly statistically significant.

29. 29 A Confidence Interval for  1 /  2

30. 30 A Confidence Interval for  1 /  2 The CI for is based on replacing F in the probability statement by the F variable (9.9) and manipulating the inequalities to isolate An interval for  1 /  2 results from taking the square root of each limit. The details are left for an exercise.