Also, suppose we set our significance level α at 0.05, so that we have only a 5% chance of making a Type I error. Since n = 15, our test statistic t* has n - 1 = 14 degrees of freedom. In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t* equaling 2.5. If the P-value is greater than \(\alpha\), do not reject the null hypothesis. If the P-value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. Set the significance level, \(\alpha\), the probability of making a Type I error to be small - 0.01, 0.05, or 0.10. Using the known distribution of the test statistic, calculate the P -value: "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?").Again, to conduct the hypothesis test for the population mean μ, we use the t-statistic \(t^*=\frac\) which follows a t-distribution with n - 1 degrees of freedom.
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