9-1 HYPOTHESIS TESTING 9-1.1 Statistical Hypotheses

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In the previous chapter we illustrated how to construct a con dence interval estimate of a parameter from sample data. However, many problems in engineering require that we decide whether to accept or reject a statement about some parameter. The statement is called a hypothesis, and the decision-making procedure about the hypothesis is called hypothesis testing. This is one of the most useful aspects of statistical inference, since many types of decision-making problems, tests, or experiments in the engineering world can be formulated as hypothesis-testing problems. Furthermore, as we will see, there is a very close connection between hypothesis testing and con dence intervals. Statistical hypothesis testing and con dence interval estimation of parameters are the fundamental methods used at the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a speci ed value. These simple comparative experiments are frequently encountered in practice and provide a good foundation for the more complex experimental design problems that we will discuss in s 13 and 14. In this chapter we discuss comparative experiments involving a single population, and our focus is on testing hypotheses concerning the parameters of the population. We now give a formal de nition of a statistical hypothesis.

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De nition A statistical hypothesis is a statement about the parameters of one or more populations.

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9-1 HYPOTHESIS TESTING

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Since we use probability distributions to represent populations, a statistical hypothesis may also be thought of as a statement about the probability distribution of a random variable. The hypothesis will usually involve one or more parameters of this distribution. For example, suppose that we are interested in the burning rate of a solid propellant used to power aircrew escape systems. Now burning rate is a random variable that can be described by a probability distribution. Suppose that our interest focuses on the mean burning rate (a parameter of this distribution). Speci cally, we are interested in deciding whether or not the mean burning rate is 50 centimeters per second. We may express this formally as H0: H1: 50 centimeters per second 50 centimeters per second (9-1)

50 centimeters per second in Equation 9-1 is called the null The statement H0: 50 centimeters per second is called the alternative hypothesis, and the statement H1: hypothesis. Since the alternative hypothesis speci es values of that could be either greater or less than 50 centimeters per second, it is called a two-sided alternative hypothesis. In some situations, we may wish to formulate a one-sided alternative hypothesis, as in H0: H1: 50 centimeters per second or 50 centimeters per second H1: 50 centimeters per second H0: 50 centimeters per second (9-2)

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It is important to remember that hypotheses are always statements about the population or distribution under study, not statements about the sample. The value of the population parameter speci ed in the null hypothesis (50 centimeters per second in the above example) is usually determined in one of three ways. First, it may result from past experience or knowledge of the process, or even from previous tests or experiments. The objective of hypothesis testing then is usually to determine whether the parameter value has changed. Second, this value may be determined from some theory or model regarding the process under study. Here the objective of hypothesis testing is to verify the theory or model. A third situation arises when the value of the population parameter results from external considerations, such as design or engineering speci cations, or from contractual obligations. In this situation, the usual objective of hypothesis testing is conformance testing. A procedure leading to a decision about a particular hypothesis is called a test of a hypothesis. Hypothesis-testing procedures rely on using the information in a random sample from the population of interest. If this information is consistent with the hypothesis, we will conclude that the hypothesis is true; however, if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false. We emphasize that the truth or falsity of a particular hypothesis can never be known with certainty, unless we can examine the entire population. This is usually impossible in most practical situations. Therefore, a hypothesis-testing procedure should be developed with the probability of reaching a wrong conclusion in mind. The structure of hypothesis-testing problems is identical in all the applications that we will consider. The null hypothesis is the hypothesis we wish to test. Rejection of the null hypothesis always leads to accepting the alternative hypothesis. In our treatment of hypothesis testing, the null hypothesis will always be stated so that it speci es an exact value of the 50 centimeters per second in Equation 9-1). The parameter (as in the statement H0: alternate hypothesis will allow the parameter to take on several values (as in the statement 50 centimeters per second in Equation 9-1). Testing the hypothesis involves taking H1: a random sample, computing a test statistic from the sample data, and then using the test statistic to make a decision about the null hypothesis.

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