,Click to Edit Master Title Style,Click to edit Master text styles,Second Level,Third Level,1,Slide,2011 Cengage Learning.All Rights Reserved.May not be scanned,copied,or duplicated,or posted to a publicly accessible website,in whole or in part.,Statistics for Businessand Economics,Anderson Sweeney Williams,Slides by,John Loucks,St.Edwards University,Statistics for Businessand Ec,Chapter 11 Inferences About Population Variances,Inference about a Population Variance,Inferences about,Two Populations Variances,Chapter 11 Inferences About P,Inferences About a Population Variance,If the sample variance is excessive,overfilling and,underfilling may be occurring even though the mean,is correct.,The mean filling weight is important,but also is the,variance of the filling weights.,Consider the production process of filling containers,with a liquid detergent product.,A variance can provide important decision-making,information.,By selecting a sample of containers,we can compute,a sample variance for the amount of detergent placed,in a container.,Inferences About a Population,Inferences About a Population Variance,Chi-Square Distribution,Interval Estimation of,2,Hypothesis Testing,Inferences About a Population,Chi-Square Distribution,We can use the chi-square distribution to develop,interval estimates and conduct hypothesis tests,about a population variance.,The sampling distribution of(,n,-1),s,2,/,2,has a chi-,square distribution whenever a simple random sample,of size,n,is selected from a normal population.,The chi-square distribution is based on sampling,from a normal population.,The,chi-square distribution,is the sum of squared,standardized normal random variables such as,(,z,1,),2,+(,z,2,),2,+(,z,3,),2,and so on.,Chi-Square Distribution We c,Examples of Sampling Distribution of(,n,-1),s,2,/,2,0,With 2 degrees,of freedom,With 5 degrees,of freedom,With 10 degrees,of freedom,Examples of Sampling Distribut,Chi-Square Distribution,For example,there is a.95 probability of obtaining a,c,2,(chi-square)value such that,We will use the notation to denote the value for the chi-square distribution that provides an area of,a,to the right of the stated value.,Chi-Square DistributionFor exa,95%of the,possible,2,values,2,0,.025,.025,Interval Estimation of,2,95%of the20.025.025Int,Interval Estimation of,2,Substituting(,n,1),s,2,/,s,2,for the,c,2,we get,Performing algebraic manipulation we get,There is a(1,a,)probability of obtaining a,c,2,value,such that,Interval Estimation of 2 Sub,Interval Estimate of a Population Variance,Interval Estimation of,2,where the,values are based on a chi-square,distribution with,n,-1 degrees of freedom and,where 1-,is the confidence coefficient.,Interval Estimate of a Populat,Interval Estimation of,Interval Estimate of a Population Standard Deviation,Taking the square root of the upper and lower,limits of the variance interval provides the confidence,interval for the population standard deviation.,Interval Estimation of Inter,Buyers Digest rates thermostats manufactured for,home temperature control.In a recent test,10,thermostats manufactured by ThermoRite were,selected and placed in a test room that was,maintained at a temperature of 68,o,F.The,temperature readings of the ten thermostats are,shown on the next slide.,Interval Estimation of,2,Example:Buyers Digest(A),Buyers Digest rates thermost,Interval Estimation of,2,We will use the 10 readings below to develop a,95%confidence interval estimate of the population,variance.,Example:Buyers Digest(A),Temperature,67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2,Thermostat,1 2 3 4 5 6 7 8 9 10,Interval Estimation of 2,Interval Estimation of,2,Selected Values from the Chi-Square Distribution Table,Our value,For,n,-1=10-1=9 d.f.and,a,=.05,Interval Estimation of 2Sele,Interval Estimation of,2,2,0,.025,Area in,Upper Tail,=.975,2.700,For,n,-1=10-1=9 d.f.and,a,=.05,Interval Estimation of 220.,Interval Estimation of,2,Selected Values from the Chi-Square Distribution Table,For,n,-1=10-1=9 d.f.and,a,=.05,Our value,Interval Estimation of 2Sele,2,0,.025,2.700,Interval Estimation of,2,n,-1=10-1=9 degrees of freedom and,a,=.05,19.023,Area in Upper,Tail=.025,20.0252.700Interval Estimatio,Sample variance,s,2,provides a point estimate of,2,.,Interval Estimation of,2,.33,2,2.33,A 95%confidence interval for the population variance is given by:,Sample variance s2 provides a,Left-Tailed Test,Hypothesis TestingAbout a Population Variance,where is the hypothesized value,for the population variance,Test Statistic,Hypotheses,Left-Tailed TestHypothesis Te,Left-Tailed Test(continued),Hypothesis TestingAbout a Population Variance,Reject,H,0,if,p,-value,a,p,-Value approach:,Critical value approach:,Rejection Rule,Reject,H,0,if,where is based on a chi-square,distribution with,n,-1 d.f.,Left-Tailed Test(continued)H,Right-Tailed Test,Hypothesis TestingAbout a Population Variance,where is the hypothesized value,for the population variance,Test Statistic,Hypotheses,Right-Tailed TestHy