9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN

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P-value approach is used, step 6 of the hypothesis-testing procedure can be modi ed. Speci cally, it is not necessary to state explicitly the critical region.

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9-2.3

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Connection between Hypothesis Tests and Con dence Intervals

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There is a close relationship between the test of a hypothesis about any parameter, say , and 2 % con dence interval for the parameter the con dence interval for . If [l, u] is a 10011 , the test of size of the hypothesis H0: H1:

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2 % CI [l, u]. As an illuswill lead to rejection of H0 if and only if 0 is not in the 100 11 tration, consider the escape system propellant problem discussed above. The null hypothesis 0.05. The 95% two-sided CI on can be calculated using 50 was rejected, using H0: Equation 8-7. This CI is 50.52 52.08. Because the value 0 50 is not included in this interval, the null hypothesis H0: 50 is rejected. Although hypothesis tests and CIs are equivalent procedures insofar as decision making or inference about is concerned, each provides somewhat different insights. For instance, the con dence interval provides a range of likely values for at a stated con dence level, whereas hypothesis testing is an easy framework for displaying the risk levels such as the P-value associated with a speci c decision. We will continue to illustrate the connection between the two procedures throughout the text.

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9-2.4

Type II Error and Choice of Sample Size

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In testing hypotheses, the analyst directly selects the type I error probability. However, the probability of type II error depends on the choice of sample size. In this section, we will show how to calculate the probability of type II error . We will also show how to select the sample size to obtain a speci ed value of . Finding the Probability of Type II Error Consider the two-sided hypothesis H0: H1: 1n

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Suppose that the null hypothesis is false and that the true value of the mean is 0. The test statistic Z0 is say, where Z0 X

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Therefore, the distribution of Z0 when H1 is true is Z0 Na

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(9-16)

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CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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Under H0 : = 0 Under H1: 0

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Figure 9-7 The distribution of Z0 under H0 and H1.

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The distribution of the test statistic Z0 under both the null hypothesis H0 and the alternate hypothesis H1 is shown in Fig. 9-7. From examining this gure, we note that if H1 is true, a type II error will be made only if z 2 Z0 z 2 where Z0 N1 1n , 12 . That is, the probability of the type II error is the probability that Z0 falls between z 2 and z 2 given that H1 is true. This probability is shown as the shaded portion of Fig. 9-7. Expressed mathematically, this probability is 1n 1n

n

(9-17)

where 1z2 denotes the probability to the left of z in the standard normal distribution. Note that Equation 9-17 was obtained by evaluating the probability that Z0 falls in the interval 3 z 2, z 2 4 when H1 is true. Furthermore, note that Equation 9-17 also holds if 0, due to the symmetry of the normal distribution. It is also possible to derive an equation similar to Equation 9-17 for a one-sided alternative hypothesis. Sample Size Formulas One may easily obtain formulas that determine the appropriate sample size to obtain a particular value of for a given and . For the two-sided alternative hypothesis, we know from Equation 9-17 that az or if 0, az

a z 1n

since 1 z 2 1n 2 0 when standard normal distribution. Then,

(9-18)

is positive. Let z be the 100 upper percentile of the 1 z 2 . From Equation 9-18 z z

9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN

n where

z 22

(9-19)

This approximation is good when 1 z 2 1n 2 is small compared to . For either of the one-sided alternative hypotheses the sample size required to produce a speci ed type II error with probability given and is