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Exercise

Posted January 29th, 2009 by mrychlik

This problem involves modifying the numerical experiment, using your favorite statistical software, to test the formula for the number of replications in a Z-test discussed in class.

In real life, you almost never know the variance of the reference population. Thus, the Z-test has mostly "academic" value, except when samples are large enough, when it does not matter whether you are using real variance or sample variance. The t-Student test is a suitable replacement for the Z-test when sample variance is used, and the sample size is small.

Determine experimentally the "power curve" for the t-Student test for various values of parameters:

  • $ \delta $ - the minimum significant difference
  • $ \sigma^2 $ - the true variance
  • $ \nu $ - the number of degrees of freedom (the same as $ r-1 $ where $ r $ is the number of replications)

You can pattern your experiment after the one for the Z-test, given here:

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Suggestions:

  • Use small numbers of replications, say 2-10.
  • Experimentally find the number of type II errors.
  • The number of type I errors $ \alpha $ can be obtained from the tables of the t-Student distribution. Use $ \alpha=0.01 $ and $ \alpha=0.05 $ only.