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where 2 2,n 1 and 2 2,n 1 are the upper and lower 100 2 percentage points of the chi1 square distribution with n 1 degrees of freedom, respectively. Figure 9-10(a) shows the critical region. The same test statistic is used for one-sided alternative hypotheses. For the one-sided hypothesis H0: H1: we would reject H0 if
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(9-28)
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whereas for the other one-sided hypothesis H0: H1:
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we would reject H0 if 9-10(b) and (c). EXAMPLE 9-8
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The one-sided critical regions are shown in Figure
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An automatic lling machine is used to ll bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of ll volume of s2 0.0153 ( uid ounces)2. If the variance of ll volume exceeds 0.01 ( uid ounces)2, an unacceptable proportion of bottles will be under lled or over lled. Is there evidence in the sample data to suggest that the manufacturer has a problem with under lled or over lled bottles Use 0.05, and assume that ll volume has a normal distribution. Using the eight-step procedure results in the following: 1. The parameter of interest is the population variance 2. H0: 2 0.01 3. H1: 2 0.01 4. 0.05 5. The test statistic is
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10-4 HYPOTHESIS TESTS ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION
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6. Reject H0 if 2 0 7. Computations:
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2 Conclusions: Since 2 29.07 30.14, we conclude that there is no 0.05,19 0 strong evidence that the variance of ll volume exceeds 0.01 ( uid ounces)2.
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Using Appendix Table III, it is easy to place bounds on the P-value of a chi-square test. From inspection of the table, we nd that 2 27.20 and 2 30.14. Since 0.10,19 0.05,19 27.20 29.07 30.14, we conclude that the P-value for the test in Example 9-8 is in the interval 0.05 P 0.10. The actual P-value is P 0.0649. (This value was obtained from a calculator.)
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Operating characteristic curves for the chi-square tests in Section 9-4.1 are provided in Appendix Charts VIi through VIn for 0.05 and 0.01. For the two-sided alternative hypothesis of Equation 9-26, Charts VIi and VIj plot against an abscissa parameter (9-30)
for various sample sizes n, where denotes the true value of the standard deviation. Charts 2 VIk and VIl are for the one-sided alternative H1: 2 0, while Charts VIm and VIn are for 2 2 the other one-sided alternative H1: as the value 0. In using these charts, we think of of the standard deviation that we want to detect. These curves can be used to evaluate the -error (or power) associated with a particular test. Alternatively, they can be used to design a test that is, to determine what sample size is necessary to detect a particular value of that differs from the hypothesized value 0. EXAMPLE 9-9 Consider the bottle- lling problem from Example 9-8. If the variance of the lling process exceeds 0.01 ( uid ounces)2, too many bottles will be under lled. Thus, the hypothesized value of the standard deviation is 0 0.10. Suppose that if the true standard deviation of the lling 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 require 0.125 0.10 1.25
This is the abscissa parameter for Chart VIk. From this chart, with n 20 and 1.25, we nd that 0.6. 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 uid ounce. To reduce the -error, a larger sample size must be used. From the operating characteristic curve with 0.20 and 1.25, we nd that n 75, approximately. Thus, if we want the test to perform as required above, the sample size must be at least 75 bottles.