Let X1, X2, ... , X21 and Y1, Y2, ... , X21 be independent random samples of sizes n =

Chapter 7, Problem 7.7-5

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Let X1, X2, ... , X21 and Y1, Y2, ... , X21 be independent random samples of sizes n = 21 and m = 21 from N(0, 1) distributions. Then F = S2 X /S2 Y has an F distribution with 20 and 20 degrees of freedom. (a) Illustrate this situation empirically by simulating 100 observations of F. (i) Plot a relative frequency histogram with the F(20, 20) pdf superimposed. (ii) Construct a qq plot of the quantiles of F(20, 20) versus the order statistics of your simulated data. Is the plot linear? (b) Consider the following 21 observations of the N(0, 1) random variable X: 0.1616 0.8593 0.3105 0.3932 0.2357 0.9697 1.3633 0.4166 0.7540 1.0570 0.1287 0.6172 0.3208 0.9637 0.2494 1.1907 2.4699 0.1931 1.2274 1.2826 1.1532 Consider also the following 21 observations of the N(0, 1) random variable Y: 0.4419 0.2313 0.9233 0.1203 1.7659 0.2022 0.9036 0.4996 0.8778 0.8574 2.7574 1.1033 0.7066 1.3595 0.0056 0.5545 0.1491 0.9774 0.0868 1.7462 0.2636 Sampling with replacement, resample with a sample of size 21 from each of these sets of observations. Calculate the value of w = s2 x/s2 y. Repeat in order to simulate 100 observations of W from these two empirical distributions. Use the same graphical comparisons that you used in part (a) to see if the 100 observations represent observations from an approximate F(20, 20) distribution. (c) Consider the following 21 observations of the exponential random variable X with mean 1: 0.6958 1.6394 0.2464 1.5827 0.0201 0.4544 0.8427 0.6385 0.1307 1.0223 1.3423 1.6653 0.0081 5.2150 0.5453 0.08440 1.2346 0.5721 1.5167 0.4843 0.9145 Consider also the following 21 observations of the exponential random variable Y with mean 1: 1.1921 0.3708 0.0874 0.5696 0.1192 0.0164 1.6482 0.2453 0.4522 3.2312 1.4745 0.8870 2.8097 0.8533 0.1466 0.9494 0.0485 4.4379 1.1244 0.2624 1.3655 Sampling with replacement, resample with a sample of size 21 from each of these sets of observations. Calculate the value of w = s2 x/s2 y. Repeat in order to simulate 100 observations of W from these two empirical distributions. Use the same graphical comparisons that you used in part (a) to see if the 100 observations represent observations from an approximate F(20, 20) distribution.