Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 9Tests of Hypotheses for a Single SampleApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9-4.1 Hypothesis Test on the Variance2Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionSuppose that we wish to test the hypothesis that the variance of a normal population 2equals a specified value, say , or equivalently, that the standard deviation is equal to 0. Let X1, X2,... ,Xnbe a random sample of n observations from this population. To test(9-6)we will use the test statistic:(9-7)20212020::HH20220)1(SnXCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9-4.1 Hypothesis Test on the Variance3Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionIf the null hypothesis is true, the test statistic defined in Equation 9-7 follows the chi-square distribution with n 1 degrees of freedom. This is the reference distribution for this test procedure. Therefore, we calculate , the value of the test statistic , and the null hypothesis would be rejected ifwhere and are the upper and lower 100 /2 percentage points of the chi-square distribution with n1 degrees of freedom, respectively. Figure 9-17(a) shows the critical region.2020: H20X20X2020: H2121ifornn21n21nCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9-4.1 Hypothesis Test on the Variance4Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionThe same test statistic is used for one-sided alternative hypotheses. For the one-sided hypotheses.(9-8)we would reject H0 if , whereas for the other one-sided hypotheses(9-9)we would reject H0if . The one-sided critical regions are shown in Fig. 9-17(b) and (c).20212020::HH21n21n 20212020::HHCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9-4.1 Hypothesis Tests on the Variance 5Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution6Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionEXAMPLE 9-8 Automated FillingAn automated filling machine is used to fill bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of fill volume of s2= 0.0153 (fluid ounces)2. If the variance of fill volume exceeds 0.01 (fluid ounces)2, an unacceptable proportion of bottles will be underfilled or overfilled. Is there evidence in the sample data to suggest that the manufacturer has a problem with underfilled or overfilled bottles? Use = 0.05, and assume that fill volume has a normal distribution.Using the seven-step procedure results in the following:1. Parameter of Interest: The parameter of interest is the population variance 2.2. Null hypothesis: H0: 2= 0.013. Alternative hypothesis: H1: 2> 0.01Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal DistributionExample 9-87Sec 9-4 Tests of the Variance & Standard Deviation of a Normal Distribution4. Test statistic: The test statistic is5. Reject H0: Use = 0.05, and reject H0if .6. Computations:7. Conclusions: Since , we conclude that there is no strong evidence that the variance of fill volume exceeds 0.01(fluid ounces)2. So there is no strong evidence of a problem with incorrectly filled bottles. 2021sn14.30207.2901.0)0153.0(1914.3007.292Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9-4.2 Type II Error and Choice of Sample SizeFor the two-sided alternative hypothesis:Operating characteristic curves are provided in Charts VIi and VIj:8Sec 9-4 Tests of the Variance & Standard Deviation of a Normal Distribution0Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution9Sec 9-4 Tests of the Variance & Standard Deviation of a Normal DistributionEXAMPLE 9-9 Automated Filling Sample SizeConsider the bottle-filling problem from Example 9-8. If the variance of the filling process exceeds 0.01 (fluid ounces)2, too many bottles will be underfilled. Thus, the hypothesized value of the standard deviation is 0= 0.10. Suppose that if the true standard deviation of the filling process exceeds this value by 25%, we would like to detect this with probability at least 0.8. Is the sample size of n = 20 adequate?To solve this problem, note that we requireThis is the abscissa parameter for Chart VIIk. From this chart, with n = 20 and = 1.25, we find that . Therefore, there is only about a 40% chance that the null hypothesis will be rejected if the true standard deviation is really as large as = 0.125 fluid ounce.To reduce the -error, a larger sample size must be used. From the operating characteristic curve with = 0.20 and = 1.25, we find that n = 75, approximately. Thus, if we want the test to perform as required above, the sample size must be at least 75 bottles.25.110.0125.006.0~Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-5 Tests on a Population Proportion9-5.1 Large-Sample Tests on a ProportionMany engineering decision problems include hypothesis testing about p.An appropriate test statistic is(9-10)10Sec 9-5 Tests on a Population Proportion0010::H p pH p p)1(0000pnpnpXZCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.9-5
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