lecture18 notes

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

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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 0 : 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 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 0 : p 1 = p 2 vs. H 1 : p 1 6 = p 2 : Z = ˆ p 1 - ˆ p 2 - E p 1 - ˆ p 2 ) p Var 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. Solution. Let X i and Y i be the first and the second midterm scores for the i -th student. Perform the t -test on H 0 : μ Y - μ X = 0
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