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fdist1 TESTING THE EQUALITY OF TWO VARIANCES: THE F TEST Application test assumption of equal variances that was made in using the t-test interest in actually comparing the variance of two populations

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fdist2 The F-Distribution Assume we repeatedly select a random sample of size n from two normal populations. Consider the distribution of the ratio of two variances:F = s 1 2 /s 1 2. The distribution formed in this manner approximates an F distribution with the following degrees of freedom: v 1 = n 1 - 1 and v 1 = n 1 - 1

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fdist3 Assumptions Random, independent samples from 2 normal populations Variability

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fdist4 F-Table The F table can be found on the appendix of our text. It gives the critical values of the F-distribution which depend upon the degrees of freedom.

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fdist5 Example 1 Assume that we have two samples with: n 2 = 7 and n 1 =10 df = 7-1= 6 and df = 10-1= 9 Let v = F(6,9) where 6 is the df from the numerator and 9 is the df of the denominator. Using the table with the appropriate df, we find : P(v < 3.37) = 0.95.

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fdist6 Example 2: Hypothesis Test to Compare Two Variances 1.Formulate the null and alternate hypotheses. H 0 : 1 2 = 1 2 H a : 1 2 > 2 2 [Note that we might also use 1 2 < 2 2 or 1 2 =/ 2 2 ] 2.Calculate the F ratio. F = s 1 2 /s 1 2 [where s 1 is the largest or the two variances] 3.Reject the null hypothesis of equal population variances if F(v 1 -1, v 2 -1) > F [or F /2 in the case of a two tailed test]

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fdist7 Example 2 The variability in the amount of impurities present in a batch of chemicals used for a particular process depends on the length of time that the process is in operation. Suppose a sample of size 25 is drawn from the normal process which is to be compared to a sample of a new process that has been developed to reduce the variability of impurities. Sample 1Sample 2 n 2525 s 2 1.040.51

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fdist8 Example 2 continued H 0 : 1 2 = 2 2 H a : 1 2 > 2 2 F(24,24) = s 1 2 /s 2 2 = 1.04/.51 = 2.04 Assuming = 0.05 cv = 1.98 < 2.04 Thus, reject H 0 and conclude that the variability in the new process (Sample 2) is less than the variability in the original process.

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fdist9 Try This A manufacturer wishes to determine whether there is less variability in the silver plating done by Company 1 than that done by Company 2. Independent random samples yield the following results. Do the populations have different variances? [solution: reject H 0 since 3.14 > 2.82]

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BASIC STATISTICAL INFERENCE A. COMPARE BETWEEN TWO MEANS OF POPULATIONS B. COMPARE BETWEEN TWO VARIANCES OF POPULATIONS PARAMETERIC TESTS (QUANTITATIVE.

BASIC STATISTICAL INFERENCE A. COMPARE BETWEEN TWO MEANS OF POPULATIONS B. COMPARE BETWEEN TWO VARIANCES OF POPULATIONS PARAMETERIC TESTS (QUANTITATIVE.

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