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Answers Chapter 16 ZandTTest.xls Answers Using two different values of n (each greater than 3) and the normal distribution for the errors, demonstrate a) that the sample slope is normally distributed b) that the test statistic, (b1 − null)/estSE, is t distributed c) that, as n increases, the test statistic, (b1 − null)estSE, gets closer and closer to the normal curve. Put your results into a Word or Excel document copying the pictures of the Monte Carlo experiments with the Take a Picture of the Results button and adding some explanatory text. a) That the sample slope is normally distributed Summary Statistics Average 4.9973 SD 0.4479 Max 6.9845 Min 3.1621 Statistic Error Box SD Errors n Normal Tails Slope Normal 1 5 10.0% Summary Statistics Average 4.9986 SD 0.2592 Max 6.0948 Min 3.9896 Empirical Histogram for 30000 Repetitions 3 4 5 6 Statistic Error Box SD Errors n Normal Tails Slope Normal 1 15 10.1% Empirical Histogram for 30000 Repetitions 7 3 sample slope estimates 4 5 6 7 sample slope estimates In each case, the normal curve has been superimposed over the sampling distribution. This is strong evidence that the sample slope is normally distributed when the errors are normal regardless of the sample size. ZandTTestAns.doc Page 1 of 2 Answers Chapter 16 b) that the test statistic, (b1 − null)/estSE, is t distributed c) that, as n increases, the test statistic, (b1 − null)estSE, gets closer and closer to the normal curve. Summary Statistics Average 0.0038 SD 1.6951 Max 24.2476 Min -18.2933 Statistic Error Box SD Errors n Normal Tails Test Stat Normal 5 5 19.9% Summary Statistics Average -0.0021 SD 1.0919 Max 6.4611 Min -6.1211 Empirical Histogram for 10000 Repetitions -10 -5 0 5 Statistic Error Box SD Errors n Normal Tails Test Stat Normal 5 15 12.4% Empirical Histogram for 10000 Repetitions 10 -10 test stat values -5 0 5 10 test stat values The two pictures above, at two values of n, demonstrate both that the t-distribution works better for smaller sample probability histograms and that the t-distribution converges to the normal as the sample size increases. On the left, we see that the normal distribution over estimates the probability near 0, whereas it underestimates the probability near the tails. The t-distribution follows the Monte Carlo histogram almost exactly. On the right, the Normal, the t, and the Monte Carlo histogram have all converged. ZandTTestAns.doc Page 2 of 2