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

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Stat 312: Lecture 16 Two sample test Moo K. Chung [email protected] November 4, 2004 1. Let X 1 , · · · , X n and Y 1 , · · · , Y m be two indepen- dent samples with mean μ X , μ Y and variance σ 2 X , σ 2 Y respectively. E ( ¯ X - ¯ Y ) = μ X - μ Y . Var ( ¯ X - ¯ Y ) = σ 2 X /n + σ 2 Y /m. 2. Let X 1 , · · · , X n and Y 1 , · · · , Y m be two inde- pendent samples with X i N ( μ X , σ 2 X ) and Y j N ( μ Y , σ 2 Y ). Then we can show that Z = ¯ Y - ¯ X - ( μ X - μ Y ) q σ 2 X /n + σ 2 Y /m N (0 , 1) . Proof. See lecture note. 3. Based on the above statistical assumptions on X i and Y j , we are interested in testing H 0 : μ X = μ Y vs. H 1 : μ X 6 = μ Y . Example 1. 12 men and 9 women were em- ployed in the experiment. Let X i be the reac- tion time of the i -th man and Y j be the re- action time of the j -th woman. There is a reason to believe that X i N ( μ X , 0 . 18) and Y j N ( μ Y , 0 . 14). The following measure- ments were taken: ¯ x = 3 . 6 seconds, ¯ y = 3 . 8 seconds. Is there any difference between the reaction times of men and women? > z=(3.6-3.8)/sqrt(0.18/12+0.14/9) > z [1] -1.144155 > qnorm(0.025) [1] -1.959964 4. Let X 1 , · · · , X n and Y 1 , · · · , Y m be two indepen- dent samples with mean
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Unformatted text preview: Assume n and m to be sufficiently large . The test statistic for testing H : μ X = μ Y vs. H 1 : μ X 6 = μ Y is Z = ¯ Y-¯ X q S 2 X /n + S 2 Y /m ∼ N (0 , 1) . Example 2. A study was conducted to compare the reaction times of men and women to a stim-ulus. 50 men and 65 women were employed in the experiment. Let X i be the reaction time of the i-th man and Y j be the reaction time of the j-th woman. The following measurements were taken: ¯ x = 3 . 6 seconds, ¯ y = 3 . 8 seconds, s 2 X = 0 . 18, s 2 Y = 0 . 14. Is there any difference between the reaction times of men and women? > z=(3.6-3.8)/sqrt(0.18/50+0.14/65) > z [1] -2.63664 What is the main reason you are getting the different results in Example 1 and 2? Review Problems. Determine the P-values in Ex-ample 1 and 2. Example 9.1., 9.4....
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