In the previous sections of this chapter we have illustrated hypothesis testing applications that are considered significance tests. After formulating the null and alternative hypotheses, we selected a sample and computed the value of a test statistic and the associated p-value. We then compared thep-value to a controlled probability of a Type I error, a, which is called the level of significance for the test. If p-value < a, we made the conclusion “reject H0” and declared the results significant; otherwise, we made the conclusion “do not reject H0.” With a significance test, we control the probability of making the Type I error, but not the Type II error. Thus, we recommended the conclusion “do not reject H0” rather than “accept H0” because the latter puts us at risk of making the Type II error of accepting H0 when it is false. With the conclusion “do not reject H0,” the statistical evidence is considered inconclusive and is usually an indication to postpone a decision or action until further research and testing can be undertaken.
However, if the purpose of a hypothesis test is to make a decision when H0 is true and a different decision when Ha is true, the decision maker may want to, and in some cases be forced to, take action with both the conclusion do not reject H0 and the conclusion reject H0. If this situation occurs, statisticians generally recommend controlling the probability of making a Type II error. With the probabilities of both the Type I and Type II error controlled, the conclusion from the hypothesis test is either to accept H0 or reject H0. In the first case, H0 is concluded to be true, while in the second case, Ha is concluded true. Thus, a decision and appropriate action can be taken when either conclusion is reached.
A good illustration of hypothesis testing for decision making is lot-acceptance sampling, a topic we will discuss in more depth in Chapter 20. For example, a quality control manager must decide to accept a shipment of batteries from a supplier or to return the shipment because of poor quality. Assume that design specifications require batteries from the supplier to have a mean useful life of at least 120 hours. To evaluate the quality of an incoming shipment, a sample of 36 batteries will be selected and tested. On the basis of the sample, a decision must be made to accept the shipment of batteries or to return it to the supplier because of poor quality. Let m denote the mean number of hours of useful life for batteries in the shipment. The null and alternative hypotheses about the population mean follow.
If H0 is rejected, the alternative hypothesis is concluded to be true. This conclusion indicates that the appropriate action is to return the shipment to the supplier. However, if H0 is not rejected, the decision maker must still determine what action should be taken. Thus, without directly concluding that H0 is true, but merely by not rejecting it, the decision maker will have made the decision to accept the shipment as being of satisfactory quality.
In such decision-making situations, it is recommended that the hypothesis testing procedure be extended to control the probability of making a Type II error. Because a decision will be made and action taken when we do not reject H0, knowledge of the probability of making a Type II error will be helpful. In Sections 9.7 and 9.8 we explain how to compute the probability of making a Type II error and how the sample size can be adjusted to help control the probability of making a Type II error.
Source: Anderson David R., Sweeney Dennis J., Williams Thomas A. (2019), Statistics for Business & Economics, Cengage Learning; 14th edition.