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 H_{0}” and declared the results significant; otherwise, we made the conclusion “do not reject H_{0}.” 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 H_{0}” rather than “accept H_{0}” because the latter puts us at risk of making the Type II error of accepting H_{0} when it is false. With the conclusion “do not reject H_{0},” 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 H_{0} is true and a different decision when H_{a} is true, the decision maker may want to, and in some cases be forced to, take action with both the conclusion do not reject H_{0} and the conclusion reject H_{0}. 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 H_{0} or reject H_{0}. In the first case, H_{0} is concluded to be true, while in the second case, H_{a} 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 H_{0} 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 H_{0} is not rejected, the decision maker must still determine what action should be taken. Thus, without directly concluding that H_{0} 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 H_{0}, 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.

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