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

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Stat 312: Lecture 14 Two-sample tests Moo K. Chung [email protected] March 11, 2003 Concepts 1. Let X 1 , · · · , X n and Y 1 , · · · , Y m be two in- dependent 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 Z = ¯ X - ¯ Y q σ 2 X /n + σ 2 Y /m N (0 , 1) . 3. Assume n and m to be large . The test statistic for testing H 0 : μ X = μ Y vs. H 1 : μ X 6 = μ Y Z = ¯ X - ¯ Y q σ 2 X /n + σ 2 Y /m N (0 , 1) . In-class problems Example 1. A study was conducted to compare the reaction times of men and women to a stim- ulus. 12 men and 9 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. There is a reason to believe that X i N ( μ X , 0 . 18) and Y j N ( μ Y , 0 . 14). We want to test if the population means are the same. The following measurements were taken: ¯ x = 3 . 6 seconds, ¯ y = 3 . 8 seconds. Is there any difference between the reaction times of men and
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Unformatted text preview: women? > z=(3.6-3.8)/sqrt(0.18/12+0.14/9) > z [1] -1.144155 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 diFerence between the reaction times of men and women? > z=(3.6-3.8)/sqrt(0.18/50+0.14/65) > z [1] -2.63664 > qnorm(0.025) [1] -1.959964 What is the main reason you are getting the diFerent results in Example 1 and 2? Self-study problems Determine the P-value in Example 1 and 2....
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