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Unformatted text preview: Course Notes for Statistics 301, Professor Wardrop Chapter 2: Two Sampling distributions The sampling distribution of the test statistic for Fisher’s test for the Chronic Crohn’s Disease study: x P ( X = x ) P ( X ≤ x ) P ( X ≥ x ) − . 46 . 0001 . 0001 1 . 0000 − . 41 . 0005 . 0006 . 9999 − . 35 . 0025 . 0031 . 9994 − . 29 . 0092 . 0123 . 9969 − . 24 . 0265 . 0388 . 9877 − . 18 . 0605 . 0993 . 9612 − . 12 . 1102 . 2095 . 9007 − . 07 . 1605 . 3700 . 7905 − . 01 . 1872 . 5572 . 6300 . 05 . 1752 . 7323 . 4428 . 10 . 1314 . 8637 . 2677 . 16 . 0788 . 9425 . 1363 . 21 . 0377 . 9802 . 0575 . 27 . 0143 . 9945 . 0198 . 33 . 0043 . 9988 . 0055 . 38 . 0010 . 9998 . 0012 . 44 . 0002 1 . 0000 . 0002 The sampling distribution of the test statistic for Fisher’s test for the Ballerina study: x P ( X = x ) P ( X ≤ x ) P ( X ≥ x ) − . 40 . 0009 . 0010 1 . 0000 − . 32 . 0081 . 0090 . 9990 − . 24 . 0387 . 0477 . 9910 − . 16 . 1127 . 1604 . 9523 − . 08 . 2104 . 3708 . 8396 . 00 . 2584 . 6292 . 6292 . 08 . 2104 . 8396 . 3708 . 16 . 1127 . 9523 . 1604 . 24 . 0387 . 9910 . 0477 . 32 . 0081 . 9990 . 0090 . 40 . 0009 1 . 0000 . 0010 The effect of sample size. Consider again the Infidelity Study. Many people find it surprising that such a large difference in ˆ p ’s is not statistically significant. As we will explore in this section, the large Pvalue is b/c of the small amount of data. To keep this simple and yet to convey the essential results, I will restrict attention to studies for which the alternative is p 1 > p 2 and the observed value of the test statistic is positive ( x > ). In the Infidelity Study, for the alternative > , the Pvalue is 0.1849. The actual Infidelity Study, of course, had n = 20 subjects. Define the imaginary doubled study version of the Infidelity Study to have twice as many subjects (40) with all values in the table doubled. This would yield the following table of data. Tell? Cheater Yes No Total ˆ p Husband 14 6 20 0.70 Wife 8 12 20 0.40 Total 22 18 40 Using the web site, the Pvalue for the alternative > is 0.0555, which is smaller than the Pvalue for the actual study. Similarly, we can define the imaginary tripled study version of the Infidelity Study to have thrice as many subjects (60) with all values in the table tripled. This would yield the following table of data. Tell? Cheater Yes No Total ˆ p Husband 21 9 30 0.70 Wife 12 18 30 0.40 Total 33 27 60 Using the web site, the Pvalue for the alternative > is 0.0185, which is smaller than both the Pvalue for the actual study and the Pvalue for the imaginary doubled study. Below is the general result. 1 Suppose that for the actual study, the alternative is > and x > . Define the imaginary ktimes study to yield the table one gets by multiplying each entry in the actual contingency table by k , where k is an integer larger than one....
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This note was uploaded on 10/23/2009 for the course STAT STATS 301 taught by Professor Professorwardrop during the Fall '08 term at University of Wisconsin.
 Fall '08
 ProfessorWardrop
 Statistics

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