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Lecture

# Comparing two means.pdf

8 Pages
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School
University of Alberta
Department
Statistics
Course
STAT368
Professor
Douglas Wiens
Semester
Winter

Description
24 3. Comparing two means  Testing whether or not certain treatments have the same e ects often involves estimating the vari- ance among the group averages, and comparing this to an estimate of the underlying, residual variation. The former is attributed to di erences between the treatments, the latter to natural vari- ation that one cannot control. Large values of the ratio of these variances indicates true treat- ment di erences. Mathematically, one computes 1 0 = 2 2 2 2 2 2 2 where  for = 1 2 and  , 1 ar2 independent r.v.s. Then 0 is the numerical value of 2 1 1 2 2 2 where the two 2s are independent; this is the denition of an r.v. on ( 1 2) degrees of freedom: 1. 2 25  Inferences involving just one mean (or the di er- ence between two means) can be based on the - distribution. One can generally reduce the prob- lem to the following. We observe 1 ³ 2´ . Then Ã ! 2 2 ¯ independently of 2 2 1 1 Here we use: 1. R.v.s 1 are jointly normally distributed if and only if every linear combination of them is also normal; this is the single most important property of the Normal distribution. 2. In Normal samples the estimate of the mean and the estimate of the variation around that mean are independent. 26  Thus = ¯ is the numerical value of the 0 ratio of independent r.v.s ¯ = q , where (0 1); 2 ( 1) 1 this is the denition of Students  on 1 d.f.  You should now be able to show that 2 follows 1 an 1distribution.  Return now to the mortar comparison problem. n o Let =1 be the samples ( = 1 for modied, = 2 for unmodied; 1 = 2 = 10). Assume: ³ ´ 11 1 1 1 ³ ´ 2 21 2 2 2 2 2 2 First assume as well that 1 = 2 = , say. Then Ã Ã !! ¯ ¯ 2 1 1 1 2 + 1 2 27 2 The pooled estimate of is 2 2 2 ( 1 1) 1+ ( 2 1) 2 = + 2 1 2 2 1 + 2 1 2 1 2 1 + 2 2 2
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