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

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Stat 312: Lecture 16 Other two sample tests Moo K. Chung [email protected] March 25, 2003 Concepts 1. Paired data: for a given paired sample ( X 1 , Y 1 ) , · · · , ( X n , Y n ) with X i = μ X and 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 X ) and Y j Bernoulli ( p Y ) . Let ˆ p X = n i =1 X i /n and ˆ p Y = m j =1 Y j /m . Var p X - ˆ p Y ) = p X (1 - p X ) n + p Y (1 - p Y ) m . 3. Difference between population proportions: for large n and m , use a Z -statistic for testing H 0 : p X = p Y : Z = ˆ p X - ˆ p Y - p X - ˆ p Y ) p Var p X - ˆ p Y ) N (0 , 1) . Since p X and p Y are unknown, we estimate them from the samples. In-class problems 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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