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UT Dallas CS 6313 - ch08-2

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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.3Let X1, X2, , Xnbe a random sample from a normal distribution with mean  and variance 2, and let S2be the sample variance. Then the random variable(8-8)has a chi-square (2) distribution with n  1 degrees of freedom. 2221SnXSec 8-3 Confidence Interval on σ2& σ of a Normal Distribution2 DistributionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Confidence Interval on the Variance and Standard Deviation4If s2is the sample variance from a random sample of nobservations from a normal distribution with unknown variance 2, then a 100(1 – )% confidence interval on 2is(8-9)where and are the upper and lower 100/2 percentage points of the chi-square distribution with n – 1 degrees of freedom, respectively. A confidence interval for  has lower and upper limits that are the square roots of the corresponding limits in Equation 8–9.1212)1()1(nnsnsn1n1nSec 8-3 Confidence Interval on σ2& σ of a Normal DistributionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.One-Sided Confidence Bounds5The 100(1 – )% lower and upper confidence bounds on 2are(8-10)2211( 1) ( 1) andnnn s n s      Sec 8-3 Confidence Interval on σ2& σ of a Normal DistributionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 8-7 Detergent Filling 6An automatic 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= 0.01532. Assume that the fill volume is approximately normal. Compute a 95% upper confidence bound.A 95% upper confidence bound is found from Equation 8-10 as follows: A confidence interval on the standard deviation  can be obtained by taking the square root on both sides, resulting inSec 8-3 Confidence Interval on σ2& σ of a Normal Distribution  2222120 1 0.015310.1170.0287ns0.17Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Normal Approximation for Binomial Proportion8-4 A Large-Sample Confidence Interval For a Population ProportionThe quantity is called the standard error of the point estimator .npp /)1( Pˆ7If n is large, the distribution ofis approximately standard normal.npppPpnpnpXZ)1(ˆ)1( Sec 8-4 Large-Sample Confidence Interval for a Population ProportionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Approximate Confidence Interval on a Binomial Proportion 8If is the proportion of observations in a random sample of size n, an approximate 100(1  )% confidence interval on the proportion p of the population is(8-11)where z/2is the upper /2 percentage point of the standard normal distribution.nppzppnppzp)ˆ1(ˆˆ)ˆ1(ˆˆSec 8-4 Large-Sample Confidence Interval for a Population ProportionˆpCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 8-8 Crankshaft Bearings 9In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish that is rougher than the specifications allow. Construct a 95% two-sided confidence interval for p.A point estimate of the proportion of bearings in the population that exceeds the roughness specification is . A 95% two-sided confidence interval for p is computed from Equation 8-11 asInterpretation: This is a wide CI. Although the sample size does not appear to be small (n = 85), the value of is fairly small, which leads to a large standard error for contributing to the wide CI.12.085/10/ˆ nxp      0.025 0.025ˆ ˆ ˆ ˆ11ˆˆ0.12 0.880.0509 0.220.12 0.880.12 1.96 0.12 1.96854385p p p pp z p p znnpp      Sec 8-4 Large-Sample Confidence Interval for a Population ProportionCopyright © 2014 John Wiley & Sons, Inc. All rights reserved.Sample size for a specified error on a binomial proportion :If we set and solve for n, the appropriate sample size isThe sample size from Equation 8-12 will always be a maximum for p = 0.5 [that is, p(1 − p) ≤ 0.25 with equality for p = 0.5], and can be used to obtain an upper bound on n.10 ppEzn 12 25.02EznSec 8-4 Large-Sample Confidence Interval for a Population ProportionChoice of Sample Size 1/E z p p n(8-12)(8-13)Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.Example 8-9 Crankshaft Bearings 11Consider the situation in Example 8-8. How large a sample is required if we want to be 95% confident that the error in using to estimate p is less than 0.05? Using as an initial estimate of p, we find from Equation 8-12 that the required sample size isIf we wanted to be at least 95% confident that our estimate of the true proportion p was within 0.05 regardless of the value of p, we would use Equation 8-13 to find the sample sizeInterpretation: If we have information concerning the value of p, either from a preliminary sample or from past experience, we


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UT Dallas CS 6313 - ch08-2

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