How to draw a conclusion when the test statistic is not in the rejection region?
What if z* is not in the rejection region? Are we going to accept the null hypothesis? The answer is no. The null hypothesis is a very strong statement. It claims that the population mean is equal to a certain number. When we fail to reject the null hypothesis, we cannot say the population mean is equal to the number under the null hypothesis. Actually, we don’t know the true population mean. If we accept the null hypothesis, we will probably commit a “type II error”. This is the error that we accept the null hypothesis when it is actually false. Since we have no idea about the size of “type II error”, it is better that we don’t accept the null hypothesis. We just conclude that we don’t have an evidence to show the alternative hypothesis is true. If we are interested in the value of true population mean, we just construct a confidence interval.
