Lecture 5_1 - F 0.01 =30.82 37.5 Same Population 1 x1 1 2 3...

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Unformatted text preview: F 0.01 =30.82 37.5 Same Population 1 x1 1 2 3 x2 x3 x1x 1 x2x 2 x3 3 Differ ent Populations Lecture 5 Notes F Distribution: The F distribution is a test statistic used in all ANOVA (Analysis of Variance) techniques. I t is also used in all multiple regression techniques. An F statistic is the ratio of two chi-square variables divided by their degrees of freedom. I n the example below, I have begun with two chi-square variables divided by their degrees of freedom. Notice, that in the second step, especially if the sample sizes were huge, that one would expect to get a value reasonably close to one. (1) (2) (3) The third step is only t rue if 1 2 were equal to 2 2 . If it were t rue, then again, one would expect to get a value close to one, especially if the sample sizes were huge. If, however, 1 2 were not equal to 2 2 , then the value one would get by calculating s 1 2 /s 2 2 would of course deviate from one. Thus a null hypothesis is associated with this test statistic, and it is 1 Ho: 1 2 = 2 2 versus Ha: 1 2 2 2 This looks like a two-tailed test procedure, but, because we always put the larger sample variance on top (call it s 1 2 ), we only employ the upper tail of the F distribution. Example: Men- Xi Xm2 Women- Xi Xw2 60 100 80 4 70 78 80 100 82 4 80 =200 =8 Df=2 Df=3 Notice, students, the sample size is very small and even though the differences in the sample variances are huge, we may not be able to reject the null hypothesis of equality (because of the small sample size, and thus large variability). For this (and only this) specific two tailed F test , the larger variance is assigned to the numerator. 2 Notice the degrees of freedom assigned to this F test, are the degrees of freedom obtaining s 1 2 for the numerator and the degrees of freedom obtaining s 2 2 for the denominator. The area to the right of the value of 37.5 is roughly estimated, from table 4, to be about 0.0075 . However, because the larger variance is always placed on top (in the numerator), we must multiply the area to the right of 37.5 by two in order to get the P-value (= 0.0075x2=0.015). So P-value is more than (=0.01) thus we do not reject H o ANOVA Analysis of Variance: We will begin our study of ANOVAs by examining a Fixed Effect Model, completely randomized design. The null hypothesis associated with this ANOVA technique might appear as follows: H o : 1 = 2 = 3 H a : The null hypothesis is not true 3 The amazing thing is that we test the above hypothesis by calculating the ratio of two variances....
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Lecture 5_1 - F 0.01 =30.82 37.5 Same Population 1 x1 1 2 3...

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