Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Unformatted text preview: Stat 312: Lecture 18 Other two sample tests Moo K. Chung [email protected] November 11, 2004 Concepts 1. Paired data. For a given paired sample ( X 1 ,Y 1 ) , ··· , ( X n ,Y n ) with E X i = μ X and E Y i = μ Y , a test statistic for testing H : μ X = μ Y can be based on one sample test. 2. Let X i ∼ Bernoulli ( p 1 ) and Y j ∼ Bernoulli ( p 2 ) . Let ˆ p 1 = ∑ n i =1 X i /n and ˆ p 2 = ∑ m j =1 Y j /m . Var (ˆ p 1- ˆ p 2 ) = p 1 (1- p 1 ) n + p 2 (1- p 2 ) m . 3. Suppose p 1 and p 2 denote the proportion of individuals in population 1 and 2 respectively. For sufficiently large n and m , we use a Z-statistic for testing H : p 1 = p 2 vs. H 1 : p 1 6 = p 2 : Z = ˆ p 1- ˆ p 2- E (ˆ p 1- ˆ p 2 ) p Var (ˆ p 1- ˆ p 2 ) . Examples Example 1. 10 students took two midterm exams. Student k 01 02 03 04 05 06 07 08 09 10 Midterm 1 k 80 75 60 90 99 60 55 85 65 70 Midterm 2 k 70 60 70 72 95 66 60 80 70 60 Is the first exam easier than the second exam? Test it at level 0.05.0....
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