406 Chapter 9 Hypothesis Testing engineer thinks that the mean for cars with the larger engine will be less than this. State the appropriate null and alternate hypotheses. H0 : �� = 25.5, H1: �� < 25.5 Answers are on page 409. A hypothesis test is like a trial The purpose of a hypothesis test is to determine how likely it is that the null hypothesis is true. The idea behind hypothesis testing is the same as the idea behind a criminal trial. At the start of a trial, the defendant is assumed to be innocent. Then the evidence is presented. If the evidence strongly indicates that the defendant is guilty, we abandon the assumption of innocence and find the defendant guilty. In a hypothesis test, the null hypothesis plays the role of the defendant. At the start of a hypothesis test, we assume that the null hypothesis is true. Then we look at the evidence, which comes from data that have been collected. If the data strongly indicate that the null hypothesis is false, we abandon our assumption that it is true and believe the alternate hypothesis instead. This is referred to as rejecting the null hypothesis. SUMMARY ∙ We begin a hypothesis test by assuming the null hypothesis to be true. ∙ If the data provide strong evidence against the null hypothesis, we reject it, and believe the alternate hypothesis. Objective 2 State conclusions to hypothesis tests Stating Conclusions If the null hypothesis is rejected, we conclude that H1 is true. We can state this conclusion by expressing H1 in words. We should not simply say ‘‘we reject the null hypothesis.’’ EXAMPLE 9.4 State a conclusion when the null hypothesis is rejected Boxes of a certain kind of cereal are labeled as containing 20 ounces. An inspector thinks that the mean weight may be less than this, so he performs a test of H0 : �� = 20 versus H1: �� < 20. He rejects the null hypothesis. State an appropriate conclusion. Solution Because the null hypothesis is rejected, we conclude that the alternate hypothesis is true. We express the alternate hypothesis in words: ‘‘We conclude that the mean weight of cereal boxes is less than 20 ounces.’’ Explain It Again The conclusion of a hypothesis test is like the verdict of a jury: Not rejecting H0 is like a jury verdict of not guilty. A not guilty verdict doesn’t mean that the defendant is innocent; it just means that the evidence wasn’t strong enough to be sure of guilt. Not rejecting H0 does not mean that H0 is true; it just means that the evidence wasn’t strong enough to reject it. If the null hypothesis is rejected, the conclusion is straightforward: We conclude that the null hypothesis is false and the alternate hypothesis is true. However, if the null hypothesis is not rejected, we do not conclude that the null hypothesis is true. In our formulation, the null hypothesis says that a parameter, such as ��, is equal to a certain value. Now we can never be sure that a parameter is exactly equal to a particular value. Therefore, we can never be sure that the null hypothesis is true. When we do not reject the null hypothesis, this just means that the evidence wasn’t strong enough to reject it. An appropriate way to state a conclusion when the null hypothesis is not rejected is to state that there is not sufficient evidence to conclude that H1 is true. SUMMARY ∙ If there is sufficient evidence to reject the null hypothesis, we conclude that the alternate hypothesis is true. ∙ If there is not sufficient evidence to reject the null hypothesis, we conclude that the null hypothesis might be true, but we never conclude that the null hypothesis is true.
navidi_monk_elementary_statistics_2e_ch7-9
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