CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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0.8 Under H0: = 50 Probability density 0.6 Under H1: = 52

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Figure 9-5 The probability of type II error when 52 and n 16.

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0 46

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50 x

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When n 16, the standard deviation of X is 1n corresponding to 48.5 and 51.5 when 52 are P 148.5 X z1 Therefore P1 5.60 Z 0.802 P1Z 0.2119 0.0000 0.2119 48.5 52 0.625 5.60 and z2

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in many practical situations we would not be as concerned with making a type II error if the mean were close to the hypothesized value. We would be much more interested in detecting large differences between the true mean and the value speci ed in the null hypothesis. The type II error probability also depends on the sample size n. Suppose that the null hypothesis is H0: 50 centimeters per second and that the true value of the mean is 52. If the sample size is increased from n 10 to n 16, the situation of Fig. 9-5 results. The normal distribution on the left is the distribution of X when the mean 50, and the normal distribution on the right is the distribution of X when 52. As shown in Fig. 9-5, the type II error probability is 51.5 when 2.5 116 522 0.625, and the z-values

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51.5 52 0.625

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Recall that when n 10 and 52, we found that 0.2643; therefore, increasing the sample size results in a decrease in the probability of type II error. The results from this section and a few other similar calculations are summarized in the following table:

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Acceptance Region 48.5 48 48.5 48 x x x x 51.5 52 51.5 52

Sample Size 10 10 16 16 0.0576 0.0114 0.0164 0.0014

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52 0.2643 0.5000 0.2119 0.5000

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50.5 0.8923 0.9705 0.9445 0.9918

9-1 HYPOTHESIS TESTING

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The results in boxes were not calculated in the text but can easily be veri ed by the reader. This display and the discussion above reveal four important points: The size of the critical region, and consequently the probability of a type I error , can always be reduced by appropriate selection of the critical values. 2. Type I and type II errors are related. A decrease in the probability of one type of error always results in an increase in the probability of the other, provided that the sample size n does not change. 3. An increase in sample size will generally reduce both and , provided that the critical values are held constant. 4. When the null hypothesis is false, increases as the true value of the parameter approaches the value hypothesized in the null hypothesis. The value of decreases as the difference between the true mean and the hypothesized value increases. 1. Generally, the analyst controls the type I error probability when he or she selects the critical values. Thus, it is usually easy for the analyst to set the type I error probability at (or near) any desired value. Since the analyst can directly control the probability of wrongly rejecting H0, we always think of rejection of the null hypothesis H0 as a strong conclusion. On the other hand, the probability of type II error is not a constant, but depends on the true value of the parameter. It also depends on the sample size that we have selected. Because the type II error probability is a function of both the sample size and the extent to which the null hypothesis H0 is false, it is customary to think of the decision to accept H0 as a weak conclusion, unless we know that is acceptably small. Therefore, rather than saying we accept H0 , we prefer the terminology fail to reject H0 . Failing to reject H0 implies that we have not found suf cient evidence to reject H0, that is, to make a strong statement. Failing to reject H0 does not necessarily mean that there is a high probability that H0 is true. It may simply mean that more data are required to reach a strong conclusion. This can have important implications for the formulation of hypotheses. An important concept that we will make use of is the power of a statistical test. De nition The power of a statistical test is the probability of rejecting the null hypothesis H0 when the alternative hypothesis is true.

The power is computed as 1 , and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power properties. For example, consider the propellant burning rate problem when we 50 centimeters per second against H1: 50 centimeters per second. are testing H0: 52. When n 10, we found that 0.2643, Suppose that the true value of the mean is 1 0.2643 0.7357 when 52. so the power of this test is 1 Power is a very descriptive and concise measure of the sensitivity of a statistical test, where by sensitivity we mean the ability of the test to detect differences. In this case, the sensitivity of the test for detecting the difference between a mean burning rate of 50 centimeters per second and 52 centimeters per second is 0.7357. That is, if the true mean is really 50 and detect this differ52 centimeters per second, this test will correctly reject H0: ence 73.57% of the time. If this value of power is judged to be too low, the analyst can increase either or the sample size n.

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