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UT Dallas CS 6313 - ch08-1

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Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 8Statistical Intervals for a Single SampleApplied Statistics and Probability for EngineersSixth EditionDouglas C. Montgomery George C. RungerCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Chapter 8 Title and Outline 28Statistical Intervals for a Single Sample8-1 Confidence Interval on the Mean of a Normal distribution, σ2Known 8-1.1 Development of the Confidence Interval & Its Properties8-1.2 Choice of Sample Size8-1.3 1-Sided Confidence Bounds8-1.4 Large-Sample Confidence Interval for μ8-2 Confidence Interval on the Mean of a Normal distribution, σ2Unknown8-2.1 t Distribution8-2.2 Confidence Interval on μ8-3 Confidence Interval on σ2& σ of a Normal Distribution8-4 Large-Sample Confidence Interval for a Population Proportion8-5 Guidelines for Constructing Confidence Intervals8-6 Tolerance & Prediction Intervals8-6.1 Prediction Interval for a Future Observation8-6.2 Tolerance Interval for a Normal DistributionCHAPTER OUTLINECopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Learning Objectives for Chapter 8After careful study of this chapter, you should be able to do the following:1. Construct confidence intervals on the mean of a normal distribution, using normal distribution or t distribution method.2. Construct confidence intervals on variance and standard deviation of normal distribution.3. Construct confidence intervals on a population proportion.4. Constructing an approximate confidence interval on a parameter.5. Prediction intervals for a future observation.6. Tolerance interval for a normal population.3Chapter 8 Learning ObjectivesCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.4A confidence interval estimate for  is an interval of the form l ≤  ≤ u, where the end-points l and u are computed from the sample data. There is a probability of 1  α of selecting a sample for which the CI will contain the true value of .The endpoints or bounds l and u are called lower- and upper-confidence limits ,and 1  α is called the confidence coefficient.Sec 8-1 Confidence Interval on the Mean of a Normal, σ2Known8-1.1 Confidence Interval and its PropertiesCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Confidence Interval on the Mean, Variance Known5If is the sample mean of a random sample of size n from a normal population with known variance 2, a 100(1  α)% CI on  is given by(8-1)where zα/2is the upper 100α/2 percentage point of the standard normal distribution.Sec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownXnzxnzx //2/2/Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.6Ten measurements of impact energy (J) on specimens of A238 steel cut at 60°C are as follows: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. The impact energy is normally distributed with  = 1J. Find a 95% CI for , the mean impact energy. The required quantities are zα/2= z0.025= 1.96, n = 10,  = l, and . The resulting 95% CI is found from Equation 8-1 as follows:Interpretation: Based on the sample data, a range of highly plausible values for mean impact energy for A238 steel at 60°C is 63.84J ≤  ≤ 65.08J46.64x08.6584.6310196.146.6410196.146.642/nzxnzxSec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownEXAMPLE 8-1 Metallic Material TransitionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.8.1.2 Sample Size for Specified Error on the Mean, Variance Known7If is used as an estimate of , we can be 100(1 − α)% confident that the error will not exceed a specified amount E when the sample size is(8-2)x|2EznSec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownxCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.EXAMPLE 8-2 Metallic Material Transition8Consider the CVN test described in Example 8-1.Determine how many specimens must be tested to ensure that the 95% CI on  for A238 steel cut at 60°C has a length of at most 1.0J. The bound on error in estimation E is one-half of the length of the CI. Use Equation 8-2 to determine n with E = 0.5,  = 1, and zα/2= 1.96. Since, n must be an integer, the required sample size is n = 16. 37.155.0196.1222/EznSec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.8-1.3 One-Sided Confidence Bounds9A 100(1 − α)% upper-confidence bound for  is(8-3)and a 100(1 − α)% lower-confidence bound for is(8-4)/x z n   lnzx /Sec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 8-3 One-Sided Confidence Bound 10The same data for impact testing from Example 8-1 are usedto construct a lower, one-sided 95% confidence interval for the mean impact energy. Recall that zα= 1.64, n = 10,  = l, and .A 100(1 − α)% lower-confidence bound for  is10xzn       Sec 8-1 Confidence Interval on the Mean of a Normal, σ2Known46.64xCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.8-1.4 A Large-Sample Confidence Interval for 11When n is large, the quantityhas an approximate standard normal distribution. Consequently,(8-5)is a large sample confidence interval for , with confidence level of approximately 100(1  ).nSX/nszxnszx2//2 Sec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 8-5 Mercury Contamination12A sample of fish was selected from 53 Florida lakes, and mercury concentration in the muscle tissue was measured (ppm). The mercury concentration values were1.2301.3300.0400.0441.2000.2700.4900.1900.8300.8100.7100.5000.4901.1600.0500.1500.1900.7701.0800.9800.6300.5600.4100.7300.5900.3400.3400.8400.5000.3400.2800.3400.7500.8700.5600.1700.1800.1900.0400.4901.1000.1600.1000.2100.8600.5200.6500.2700.9400.4000.4300.2500.270Sec 8-1 Confidence Interval on the Mean of a Normal, σ2KnownFind an approximate 95% CI on .Copyright ©


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