Course Notes for Statistics 301 Professor Wardrop Chapter 2 Two Sampling distributions Chapter 3 Simulation The sampling distribution of the test statistic for Results of a simulation experiment with 10 000 Fisher s test for the Chronic Crohn s Disease study runs for the Ballerina study x P X x P X x P X x 0 46 0 0001 0 0001 1 0000 0 41 0 0005 0 0006 0 9999 0 35 0 0025 0 0031 0 9994 0 29 0 0092 0 0123 0 9969 0 24 0 0265 0 0388 0 9877 0 18 0 0605 0 0993 0 9612 0 12 0 1102 0 2095 0 9007 0 07 0 1605 0 3700 0 7905 0 01 0 1872 0 5572 0 6300 0 05 0 1752 0 7323 0 4428 0 10 0 1314 0 8637 0 2677 0 16 0 0788 0 9425 0 1363 0 21 0 0377 0 9802 0 0575 0 27 0 0143 0 9945 0 0198 0 33 0 0043 0 9988 0 0055 0 38 0 0010 0 9998 0 0012 0 44 0 0002 1 0000 0 0002 x 0 40 0 32 0 24 0 16 0 08 0 00 0 08 0 16 0 24 0 32 0 40 Rel Freq Rel Freq Rel Freq of x of x of x 0 0009 0 0009 1 0000 0 0072 0 0081 0 9991 0 0383 0 0464 0 9919 0 1137 0 1601 0 9536 0 2169 0 3770 0 8399 0 2591 0 6361 0 6230 0 2022 0 8383 0 3639 0 1140 0 9523 0 1617 0 0383 0 9906 0 0477 0 0089 0 9995 0 0094 0 0005 1 0000 0 0005 Results of a simulation experiment with 10 000 runs for the Crohn s study x 0 46 0 41 0 35 0 29 0 24 0 18 0 12 0 07 0 01 0 05 0 10 0 16 0 21 0 27 0 33 0 38 0 44 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 0 40 0 0009 0 0010 1 0000 0 32 0 0081 0 0090 0 9990 0 24 0 0387 0 0477 0 9910 0 16 0 1127 0 1604 0 9523 0 08 0 2104 0 3708 0 8396 0 00 0 2584 0 6292 0 6292 0 08 0 2104 0 8396 0 3708 0 16 0 1127 0 9523 0 1604 0 24 0 0387 0 9910 0 0477 0 32 0 0081 0 9990 0 0090 0 40 0 0009 1 0000 0 0010 1 Rel Freq Rel Freq Rel Freq of x of x of x 0 0002 0 0002 1 0000 0 0005 0 0007 0 9998 0 0027 0 0034 0 9993 0 0094 0 0128 0 9966 0 0289 0 0417 0 9872 0 0593 0 1010 0 9583 0 1178 0 2188 0 8990 0 1540 0 3728 0 7812 0 1893 0 5621 0 6272 0 1724 0 7345 0 4379 0 1287 0 8632 0 2655 0 0830 0 9462 0 1368 0 0345 0 9807 0 0538 0 0143 0 9950 0 0193 0 0039 0 9989 0 0050 0 0010 0 9999 0 0011 0 0001 1 0000 0 0001 Standard Normal Curve Approximation Recall that x p 1 p 2 and s m1 m2 n1 n2 n 1 Without the continuity correction With the continuity correction 1 Compute z x Compute 2 For the g 2 First alternative the approximate Pvalue equals the area under the standard normal curve to the right of z n 2n1 n2 For the first alternative 1 Compute x1 x g and z1 x1 Second alternative the approximate Pvalue equals the area under the standard normal curve to the right of z 2 The approximate P value equals the area under the standard normal curve to the right of z1 Third alternative 6 the approximate Pvalue equals twice the area under the standard normal curve to the right of z For the second alternative 1 Compute x2 x g and z2 x2 2 The approximate P value equals the area under the standard normal curve to the right of z2 For the third alternative 6 if x g then the exact P value equals one and no approximation is needed otherwise 1 Compute x3 x g and z3 x3 Note that both x3 and z3 are larger than zero 2 The approximate P value equals twice the area under the standard normal curve to the right of z3 2 Chapter 5 Sampling With or Without Replacement Two probability histograms for X the number of successes in a sample of size n 10 from a dichotomous box with N 1 000 and p 0 6 Solid Dashed rectangles are for a random sample with without replacement Simulation 10 000 runs Exact 0 25 0 25 0 20 0 20 0 15 0 15 0 10 0 10 0 05 0 05 2 6 4 8 10 2 6 4 8 10 Two probability histograms for X the number of successes in a sample of size n 10 from a dichotomous box with N 20 and p 0 6 Solid Dashed rectangles are for a random sample with without replacement Simulation 10 000 runs Exact 0 35 0 35 0 30 0 30 0 25 0 25 0 20 0 20 0 15 0 15 0 10 0 10 0 05 0 05 2 4 6 8 10 2 3 4 6 8 10 Chapter 7 Background lurking variables A company with 200 employees decides it must reduce its work force by one half The following table reveals the relationship between gender and outcome Gender Female Male Total Outcome Released Not released 60 40 40 60 100 100 p 0 60 0 40 Total 100 100 200 In an observational study we should consider the possible influence of a background variable For illustration suppose that the company has workers in two job classifications A and B Consider four scenarios Case 1 Gender Female Male Total Job A Outcome Released Not released 30 20 20 30 50 50 Gender Female Male Total Job A Outcome Released Not released 30 10 30 30 60 40 Gender Female Male Total Job A Outcome Released Not released 60 15 40 10 100 25 Total 50 50 100 p 0 60 0 40 Job B Outcome Gender Released Not released Female 30 20 Male 20 30 Total 50 50 Total 50 50 100 p 0 60 0 40 Job B Outcome Gender Released Not released Female 30 30 Male 10 30 Total 40 60 Total 60 40 100 p 0 50 0 25 Not released 25 50 75 Total 25 50 75 p 0 00 0 00 Job B Outcome Gender Released Not released Female 4 16 Male 24 56 Total 28 72 Total 20 80 100 p 0 20 0 30 Case 2 Total 40 60 100 p 0 75 0 50 Case 3 Job B Total 75 50 125 …