ch8 - 191 Inference about more than Two Population Central...

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Unformatted text preview: 191 Inference about more than Two Population Central Values In Chapter 5 we studied inferences about a population mean μ using a single sample from the population. In Chapter 6 we studied inferences about the difference between two population means μ 1- μ 2 based on a ran- dom sample from each population. Chapter 8 deals with inferences about means μ 1 , μ 2 , . . . , μ t from t > 2 populations. The first and foremost question that we ask is whether all the populations have the same mean? If this is the case then no other infer- ences about relationships among the set are relevant. A natural approach to think of when faced with the task of determining whether evidence points to μ 1 = μ 2 = μ 3 = ··· = μ t is to draw on knowledge acquired in Chap- ter 6 and make a t-test of H : μ i = μ j for each possible combination of i and j . If all test fail to reject then one can conclude that all means μ i are equal. This procedure has a serious flaw which willnow be described. Take the case t = 3, so that we have three populations with means μ 1 , μ 2 , μ 3 . A sample from each population yields sample means and variances ¯ y 1 , ¯ y 2 , ¯ y 3 , s 2 1 , s 2 2 , s 2 3 . All possible differences μ i- μ j are: μ 1- μ 2 , μ 1- μ 3 , μ 2- μ 3 . Testing each H : μ i- μ j = 0 will employ a t-test of the 192 form t ij = ¯ y i- ¯ y j s p v u u t 1 n i + 1 n j , s 2 p = ( n i- 1) s 2 i + ( n j- 1) s 2 j n i + n j- 2 Suppose each test is made at the same level α . Thus for each pair ( i, j ) P ( | t ij | > t α/ 2 | when H is true) = α . In other words, the Type I error probability is α for each test. We shall call α the per-comparison error rate . Now let us ask the question – What is the probability of making one or more Type I errors when we make all three tests? ( We shall call this the overall error rate. ) Theory says that if the three t random variables were in- dependent then the overall error rate is 1- (1- α ) 3 [that is 0.14 for α = 0.05] The three test statistics are not independent. For exam- ple t 12 = ¯ y 1- ¯ y 2 s p s 1 n 1 + 1 n 2 and t 13 = ¯y 1- ¯y 3 s p s 1 n 1 + 1 n 3 These two statistics both involve ¯ y 1 so they are not inde- pendent. Thus the overall error rate is not exactly 0.14 for our three test situation, and computing its exact value is difficult in theory. However suffice it to say that the overall error rate is much larger than α . In general, for c tests made at level α , the overall error 193 rate is 1- (1- α ) c if the tests are independent, and is something much larger than α if the tests are dependent....
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ch8 - 191 Inference about more than Two Population Central...

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