Determine the critical value rule for deciding whether to reject H0
Determine the critical value rule for deciding whether to reject H0. To decide how large the test statistic z must be to reject H0 in favor of Ha by setting the probability of a Type I error equal to α, we note that different samples would give different sample means and thus different values of z. Because the sample size n = 40 is large, the Central Limit Theorem tells us that the sampling distribution of z is (approximately) a standard normal distribution if the null hypothesis H0: μ = 50 is true. Therefore, we do the following:
Place the probability of a Type I error, α, in the right-hand tail of the standard normal curve and use the normal table (see Table A.3, page 863) to find the normal point zα. Here zα, which we call a critical value, is the point on the horizontal axis under the standard normal curve that gives a right-hand tail area equal to α.
Reject H0: μ = 50 in favor of Ha: μ > 50 if and only if the test statistic z is greater than the critical value zα (This is the critical value rule.)
Figure 9.1 illustrates that since we have set α equal to .05, we should use the critical value zα = z.05 = 1.645 (see Table A.3). This says that we should reject H0 if z > 1.645 and we should not reject H0 if z ≤ 1.645.
Figure 9.1: The Critical Value for Testing H0: μ = 50 versus Ha: μ > 50 by Setting α = .05
To better understand the critical value rule, consider the standard normal curve in Figure 9.1. The area of .05 in the right-hand tail of this curve implies that values of the test statistic z that are greater than 1.645 are unlikely to occur if the null hypothesis H0: μ = 50 is true. There is a 5 percent chance of observing one of these values—and thus wrongly rejecting H0—if H0 is true. However, we are more likely to observe a value of z greater than 1.645—and thus correctly reject H0—if H0 is false. Therefore, it is intuitively reasonable to reject H0 if the value of the test statistic z is greater than 1.645.