Chapter 11 Statistical Inferences for Population Variances

Statistical Inferences for Population VariancesChapter 11Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinStatistical Inferences for Population Variances11.1 The Chi-Square Distribution11.2 Statistical Inference for a Population Variance11.3 The F Distribution11.4 Comparing Two Population Variances by Using Independent Samples11-*11.1 The Chi-Square DistributionSometimes make inferences using the chi-square distributionDenoted ²Skewed to the rightExact shape depends on the degrees of freedomDenoted dfA chi-square point ²α is the point under a chi-square distribution that gives right-hand tail area LO11-1: Describe the properties of the chi-square distribution and use a chi-square table.11-*11.1 Statistical Inference for Population VarianceIf s2 is the variance of a random sample of n measurements from a normal population with variance σ2The sampling distribution of the statistic(n - 1) s2 / σ2 is a chi-square distribution with (n – 1) degrees of freedomCan calculate confidence interval and perform hypothesis testing100(1-α)% confidence interval for σ2LO11-2: Use the chi-square distribution to make statistical inferences about population variances.11-*FormulasLO11-211-*11.3 F DistributionLO11-3: Describe the properties of the F distribution and use on F table.Figure 11.511-*F Distribution TablesThe F point F is the point on the horizontal axis under the curve of the F distribution that gives a right-hand tail area equal to The value of F depends on a (the size of the right-hand tail area) and df1 and df2Different F tables for different values of Tables A.6 for  = 0.10Tables A.7 for  = 0.05Tables A.8 for  = 0.025Tables A.9 for  = 0.01LO11-311-*11.4 Comparing Two Population Variances by Using Independent SamplesPopulation 1 has variance σ12 and population 2 has variance σ22The null hypothesis H0 is that the variances are the sameH0: σ12 = σ22The alternative is that one is smaller than the otherThat population has less variable measurementsSuppose σ12 > σ22More usual to normalizeTest H0: σ12/σ22 = 1 vs. σ12/σ22 > 1LO11-4: Compare two population variances when the samples are independent.11-*Comparing Two Population Variances ContinuedReject H0 in favor of Ha if s12/s22 is significantly greater than 1s12 is the variance of a random of size n1 from a population with variance σ12 s22 is the variance of a random of size n2 from a population with variance σ22To decide how large s12/s22 must be to reject H0, describe the sampling distribution of s12/s22The sampling distribution of s12/s22 is the F distributionLO11-411-*