notes-11-22-24

notes-11-22-24 - M4056 Analysis of Variance, I November...

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Unformatted text preview: M4056 Analysis of Variance, I November 22-24, 2010 1. Introduction and Goal Let X be a normal random variable with mean X and variance 2 . Let Y be another normal random variable with mean Y and the same variance 2 as X . In the lectures of November 17 and 19, we examined how to test the hypothesis H : X = Y using the evidence obtained from a sample ( X 1 ,...,X n ) from the X distribution and a sample ( Y 1 ,...,Y m ) form the Y distribution. The key technical result that makes this possible is the fact that t = ( X Y ) ( X Y ) S p radicalbig 1 /n + 1 /m is a t distribution with m + n 2 degrees of freedom. Now suppose that we have several normal random variables Y 1 ,...,Y m . We shall assume they all have the same variance 2 . The means may be different. Let i be the mean of Y i , i { 1 ,...,m } . Let be the average of the i . In summary, Y i normal( i , 2 ) for i = 1 ,...m = 1 m m summationdisplay i =1 i . From each distribution, we take a sample ( Y i 1 ,...,Y in ). Thus, we have an m n matrix of independent random variables: Y 11 Y 12 Y 1 n Y 11 Y 12 Y 1 n . . . . . . . . . . . . Y m 1 Y m 2 Y mn Here, the i th row is i.i.d. Y i . Our goal is to devise a test for the null hypothesis: H : 1 = 2 = = m . For example, suppose m different treatments were applied to m different groups. To determine if there is any evidence that any of the treatments are effective, this is the hypothesis we would test. A significant violation of the null hypothesis would count as evidence of that at least one group was affected differently than the others. Note that in the scenario we are imagining, we have m samples, each of size n . A more general situation arises if the samples have different sizes, but we will delay consideration of this till later.arises if the samples have different sizes, but we will delay consideration of this till later....
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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notes-11-22-24 - M4056 Analysis of Variance, I November...

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