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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com 8.9 Confidence Intervals for σ 2 435 FIGURE 8.11 Location of χ 2 1 − (α/2) and χ 2 α/2 2 2 0 2 L 2 U and a reordering of the inequality in the probability statement gives [ ] (n − 1)S 2 P ≤ σ 2 (n − 1)S2 ≤ = 1 − α. χ 2 (α/2) The confidence interval for σ 2 is as follows. χ 2 1−(α/2) A 100(1 − α)% Confidence Interval for σ 2 ( (n − 1)S 2 , χ 2 α/2 ) (n − 1)S2 χ 2 1−(α/2) EXAMPLE 8.13 Solution An experimenter wanted to check the variability of measurements obtained by using equipment designed to measure the volume of an audio source. Three independent measurements recorded by this equipment for the same sound were 4.1, 5.2, and 10.2. Estimate σ 2 with confidence coefficient .90. If normality of the measurements recorded by this equipment can be assumed, the confidence interval just developed applies. For the data given, s 2 = 10.57. With α/2 = .05 and (n − 1) = 2 df, Table 6, Appendix 3, gives χ.95 2 2 = .103 and χ.05 = 5.991. Thus, the 90% confidence interval for σ 2 is ( (n − 1)s 2 ) ( (n − 1)s2 (2)(10.57) χ.05 2 , χ.95 2 or , (2)(10.57) ) , 5.991 .103 and finally, (3.53, 205.24). Notice that this interval for σ 2 is very wide, primarily because n is quite small. We have previously indicated that the confidence intervals developed in Section 8.8 for µ and µ 1 − µ 2 had confidence coefficients near the nominal level even if the underlying populations were not normally distributed. In contrast, the intervals for σ 2 presented in this section can have confidence coefficients that differ markedly from the nominal level if the sampled population is not normally distributed.

www.downloadslide.com 436 Chapter 8 Estimation Exercises 8.95 The EPA has set a maximum noise level for heavy trucks at 83 decibels (dB). The manner in which this limit is applied will greatly affect the trucking industry and the public. One way to apply the limit is to require all trucks to conform to the noise limit. A second but less satisfactory method is to require the truck fleet’s mean noise level to be less than the limit. If the latter rule is adopted, variation in the noise level from truck to truck becomes important because a large value of σ 2 would imply that many trucks exceed the limit, even if the mean fleet level were 83 dB. A random sample of six heavy trucks produced the following noise levels (in decibels): 85.4 86.8 86.1 85.3 84.8 86.0. Use these data to construct a 90% confidence interval for σ 2 , the variance of the truck noiseemission readings. Interpret your results. 8.96 In Exercise 8.81, we gave the carapace lengths of ten mature Thenus orientalis lobsters caught in the seas in the vicinity of Singapore. For your convenience, the data are reproduced here. Suppose that you wished to describe the variability of the carapace lengths of this population of lobsters. Find a 90% confidence interval for the population variance σ 2 . Lobster Field Number A061 A062 A066 A070 A067 A069 A064 A068 A065 A063 Carapace Length (mm) 78 66 65 63 60 60 58 56 52 50 8.97 Suppose that S 2 is the sample variance based on a sample of size n from a normal population with unknown mean and variance. Derive a 100(1 − α)% a upper confidence bound for σ 2 . b lower confidence bound for σ 2 . 8.98 Given a random sample of size n from a normal population with unknown mean and variance, we developed a confidence interval for the population variance σ 2 in this section. What is the formula for a confidence interval for the population standard deviation σ ? 8.99 In Exercise 8.97, you derived upper and lower confidence bounds, each with confidence coefficient 1 − α, for σ 2 . How would you construct a 100(1 − α)% a upper confidence bound for σ ? b lower confidence bound for σ ? 8.100 Industrial light bulbs should have a mean life length acceptable to potential users and a relatively small variation in life length. If some bulbs fail too early in their life, users become annoyed and are likely to switch to bulbs produced by a different manufacturer. Large variations above the mean reduce replacement sales; in general, variation in life lengths disrupts the user’s replacement schedules. A random sample of 20 bulbs produced by a particular manufacturer produced the following lengths of life (in hours): 2100 2302 1951 2067 2415 1883 2101 2146 2278 2019 1924 2183 2077 2392 2286 2501 1946 2161 2253 1827 Set up a 99% upper confidence bound for the standard deviation of the lengths of life for the bulbs produced by this manufacturer. Is the true population standard deviation less than 150 hours? Why or why not? 8.101 In laboratory work, it is desirable to run careful checks on the variability of readings produced on standard samples. In a study of the amount of calcium in drinking water undertaken as part of a water quality assessment, the same standard sample was run through the laboratory six

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