C-6e. The Relationship between Confidence Intervals and Hypothesis Testing Because constructing confidence intervals and hypothesis tests both involve probability statements, it is natural to think that they are somehow linked. It turns out that they are. After a confidence interval has been constructed, we can carry out a variety of hypothesis tests. The confidence intervals we have discussed are all two-sided by nature. (In this text, we will have no need to construct one-sided confidence intervals.) Thus, confidence intervals can be used to test against two-sided alternatives. In the case of a population mean, the null is given by (C.31), and the alternative is (C.34). Suppose we have constructed a 95% confidence interval for . Then, if the hypothesized value of under , , is not in the confidence interval, then is rejected against at the 5% level. If lies in this interval, then we fail to reject at the 5% level. Notice how any value for can be tested once a confidence interval is constructed, and because a confidence interval contains more than one value, there are many null hypotheses that will not be rejected. .... take notes and explain cleartly and easy to understand