CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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One-Sided and Two-Sided Hypotheses

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A test of any hypothesis such as H0: H1:

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is called a two-sided test, because it is important to detect differences from the hypothesized value of the mean 0 that lie on either side of 0 . In such a test, the critical region is split into two parts, with (usually) equal probability placed in each tail of the distribution of the test statistic. Many hypothesis-testing problems naturally involve a one-sided alternative hypothesis, such as H0: H1:

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H0: H1:

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If the alternative hypothesis is H1: 0 , the critical region should lie in the upper tail of the distribution of the test statistic, whereas if the alternative hypothesis is H1: 0, the critical region should lie in the lower tail of the distribution. Consequently, these tests are sometimes called one-tailed tests. The location of the critical region for one-sided tests is usually easy to determine. Simply visualize the behavior of the test statistic if the null hypothesis is true and place the critical region in the appropriate end or tail of the distribution. Generally, the inequality in the alternative hypothesis points in the direction of the critical region. In constructing hypotheses, we will always state the null hypothesis as an equality so that the probability of type I error can be controlled at a speci c value. The alternative hypothesis might be either one-sided or two-sided, depending on the conclusion to be drawn if H0 is rejected. If the objective is to make a claim involving statements such as greater than, less than, superior to, exceeds, at least, and so forth, a one-sided alternative is appropriate. If no direction is implied by the claim, or if the claim not equal to is to be made, a two-sided alternative should be used. EXAMPLE 9-1 Consider the propellant burning rate problem. Suppose that if the burning rate is less than 50 centimeters per second, we wish to show this with a strong conclusion. The hypotheses should be stated as H0: H1: 50 centimeters per second 50 centimeters per second

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Here the critical region lies in the lower tail of the distribution of X . Since the rejection of H0 is always a strong conclusion, this statement of the hypotheses will produce the desired outcome if H0 is rejected. Notice that, although the null hypothesis is stated with an equal sign, it is understood to include any value of not speci ed by the alternative hypothesis. Therefore, failing to reject H0 does not mean that 50 centimeters per second exactly, but only that we do not have strong evidence in support of H1. In some real-world problems where one-sided test procedures are indicated, it is occasionally dif cult to choose an appropriate formulation of the alternative hypothesis. For example, suppose that a soft-drink beverage bottler purchases 10-ounce bottles from a glass

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9-1 HYPOTHESIS TESTING

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company. The bottler wants to be sure that the bottles meet the speci cation on mean internal pressure or bursting strength, which for 10-ounce bottles is a minimum strength of 200 psi. The bottler has decided to formulate the decision procedure for a speci c lot of bottles as a hypothesis testing problem. There are two possible formulations for this problem, either H0: H1: or H0: H1: 200 psi 200 psi (9-6) 200 psi 200 psi

(9-5)

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Consider the formulation in Equation 9-5. If the null hypothesis is rejected, the bottles will be judged satisfactory; if H0 is not rejected, the implication is that the bottles do not conform to speci cations and should not be used. Because rejecting H0 is a strong conclusion, this formulation forces the bottle manufacturer to demonstrate that the mean bursting strength of the bottles exceeds the speci cation. Now consider the formulation in Equation 9-6. In this situation, the bottles will be judged satisfactory unless H0 is rejected. That is, we conclude that the bottles are satisfactory unless there is strong evidence to the contrary. Which formulation is correct, the one of Equation 9-5 or Equation 9-6 The answer is it depends. For Equation 9-5, there is some probability that H0 will not be rejected (i.e., we would decide that the bottles are not satisfactory), even though the true mean is slightly greater than 200 psi. This formulation implies that we want the bottle manufacturer to demonstrate that the product meets or exceeds our speci cations. Such a formulation could be appropriate if the manufacturer has experienced dif culty in meeting speci cations in the past or if product safety considerations force us to hold tightly to the 200 psi speci cation. On the other hand, for the formulation of Equation 9-6 there is some probability that H0 will be accepted and the bottles judged satisfactory, even though the true mean is slightly less than 200 psi. We would conclude that the bottles are unsatisfactory only when there is strong evi200 psi is rejected. This dence that the mean does not exceed 200 psi, that is, when H0: formulation assumes that we are relatively happy with the bottle manufacturer s past per200 psi are not harmful. formance and that small deviations from the speci cation of In formulating one-sided alternative hypotheses, we should remember that rejecting H0 is always a strong conclusion. Consequently, we should put the statement about which it is important to make a strong conclusion in the alternative hypothesis. In real-world problems, this will often depend on our point of view and experience with the situation.

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9-1.4

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