404 Chapter 9 Hypothesis Testing If there were no temperature trend, we would expect that the proportion of days on which the high temperature record was more recent to be about one-half. It would not be surprising if this proportion were somewhat different from one-half, because we would expect to see some difference just by chance. However, we would not expect the proportion to be much different from one-half. If the proportion were much different from one-half, we would conclude that there was a temperature trend. There are 26 days in the table. Half of 26 is 13. If the record high had been more recent on 13 of the days, there would be no reason to believe there was a warming trend. If the record high had been more recent on 14 or 15 days, this would be slightly more than one-half, but it would seem reasonable to believe that a difference this small was just due to chance. In fact, the record high temperature was more recent for 18 of the 26 days. The question we need to address is whether this difference from one-half is too large for us to believe that it is simply due to chance. This is the sort of question that hypothesis tests are designed to answer. In this chapter, we will learn to perform hypothesis tests in a variety of commonly occurring situations. In the case study at the end of the chapter, we will study a data set that includes the data in the preceding table, and investigate the possibility of a warming trend in Washington, D.C. SECTION 9.1 Basic Principles of Hypothesis Testing Objectives 1. Define the null and alternate hypotheses 2. State conclusions to hypothesis tests 3. Distinguish between Type I and Type II errors Objective 1 Define the null and alternate hypotheses The Null Hypothesis and the Alternate Hypothesis Air pollution has become a serious health problem in many cities. One of the forms of air pollution that health officials are most concerned about is particulate matter (PM), which refers to fine particles that can be trapped in the lungs, increasing the risk of respiratory disease. Some of the PM in the atmosphere comes from car exhaust, so one important way to reduce PM pollution is to design automobile engines that produce less PM. The following example will show how hypothesis testing can play a part in this effort. A study published in the Journal of the Air and Waste Management Association reported that the mean amount of PM produced by cars and light trucks in an urban setting is 35 milligrams of PM per mile of travel. Suppose that a new engine design is proposed that is intended to reduce this level. Now there are two possibilities: either the new design will reduce the level, or it will not. These possibilities are called hypotheses. To be specific, 1. The null hypothesis says that the new design will not reduce the level, so the mean for the new engines will be �� = 35. 2. The alternate hypothesis says that the new design will reduce the level, so �� < 35. In general, the null hypothesis says that a parameter is equal to a certain value, while the alternate hypothesis says that the parameter differs from this value. Often the null hypothesis is a statement of no change or no difference, while the alternate hypothesis states that a change or difference has occurred. DEFINITION ∙ The null hypothesis about a parameter states that the parameter is equal to a specific value, for example, H0 : �� = 35. The null hypothesis is denoted H0. ∙ The alternate hypothesis about a parameter states that the value of the parameter differs from the value specified by the null hypothesis. The alternate hypothesis is denoted H1.
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