Visually, the rejection region is shaded red in the graph. That is, we would reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3 if the test statistic t* is greater than 1.7613. It can be shown using either statistical software or a t-table that the critical value t 0.05,14 is 1.7613. The critical value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the t-value, denoted t \(\alpha\), n - 1, such that the probability to the right of it is \(\alpha\). 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 we take a random sample of n = 15 students majoring in mathematics. If the test statistic is less extreme than the critical value, do not reject the null hypothesis. If the test statistic is more extreme in the direction of the alternative than the critical value, reject the null hypothesis in favor of the alternative hypothesis. Compare the test statistic to the critical value.Determine the critical value by finding the value of the known distribution of the test statistic such that the probability of making a Type I error - which is denoted \(\alpha\) (greek letter "alpha") and is called the " significance level of the test" - is small (typically 0.01, 0.05, or 0.10).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. Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic.Specify the null and alternative hypotheses.Specifically, the four steps involved in using the critical value approach to conducting any hypothesis test are: If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected. That is, it entails comparing the observed test statistic to some cutoff value, called the " critical value." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis.
The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true.