Using confidence intervals to test hypotheses
Confidence intervals can be used to test hypotheses. Specifically, it can be proven that we can reject H0: μ = μ0 in favor of Ha: μ ≠ μ0 by setting the probability of a Type I error equal to α if and only if the 100(1 − α) percent confidence interval for μ does not contain μ0. For example, consider the Valentine’s Day chocolate case and testing H0: μ = 330 versus Ha: μ ≠ 330 by setting α equal to .05. To do this, we use the mean of the sample of n = 100 reported order quantities to calculate the 95 percent confidence interval for μ to be
Because this interval does contain 330, we cannot reject H0: μ = 330 in favor of Ha: μ ≠ 330 by setting α equal to .05.
Whereas we can use two-sided confidence intervals to test “not equal to” alternative hypotheses, we must use one-sided confidence intervals to test “greater than” or “less than” alternative hypotheses. We will not study one-sided confidence intervals in this book. However, it should be emphasized that we do not need to use confidence intervals (one-sided or two-sided) to test hypotheses. We can test hypotheses by using test statistics and critical values or p-values, and these are the approaches that we will feature throughout this book.