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,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 12 Tests of Goodness of Fit and Independence,Goodness of Fit Test:A Multinomial Population,Goodness of Fit Test:,Poisson and Normal Distributions,Test of Independence,Chapter 12 Tests of Goodness,Hypothesis(Goodness of Fit)Testfor Proportions of a Multinomial Population,1.,State the null and alternative hypotheses.,H,0,:The population follows a multinomial,distribution with specified probabilities,for each of the,k,categories,H,a,:The population does,not,follow a,multinomial distribution with specified,probabilities for each of the,k,categories,Hypothesis(Goodness of Fit)T,Hypothesis(Goodness of Fit)Testfor Proportions of a Multinomial Population,2.,Select a random sample and record the observed,frequency,f,i,for each of the,k,categories.,3.,Assuming,H,0,is true,compute the expected,frequency,e,i,in each category by multiplying the,category probability by the sample size.,Hypothesis(Goodness of Fit)T,Hypothesis(Goodness of Fit)Testfor Proportions of a Multinomial Population,4.,Compute the value of the test statistic.,Note:The test statistic has a chi-square distribution,with,k,1 df provided that the expected frequencies,are 5 or more for all categories.,f,i,=observed frequency for category,i,e,i,=expected frequency for category,i,k,=number of categories,where:,Hypothesis(Goodness of Fit)T,Hypothesis(Goodness of Fit)Testfor Proportions of a Multinomial Population,where,is the significance level and,there are,k,-1 degrees of freedom,p,-value approach:,Critical value approach:,Reject,H,0,if,p,-value,a,5.,Rejection rule:,Reject,H,0,if,Hypothesis(Goodness of Fit)T,Multinomial Distribution Goodness of Fit Test,Example:Finger Lakes Homes(A),Finger Lakes Homes manufactures four models of,prefabricated homes,a two-story colonial,a log cabin,a split-level,and an A-frame.To help in production,planning,management would like to determine if,previous customer purchases indicate that there is a,preference in the style selected.,Multinomial Distribution Goodn,Split-A-,Model Colonial Log Level Frame,#Sold,30 20 35 15,The number of homes sold of each model for 100,sales over the past two years is shown below.,Multinomial Distribution Goodness of Fit Test,Example:Finger Lakes Homes(A),Hypotheses,Multinomial Distribution Goodness of Fit Test,where:,p,C,=population proportion that purchase a colonial,p,L,=population proportion that purchase a log cabin,p,S,=population proportion that purchase a split-level,p,A,=population proportion that purchase an A-frame,H,0,:,p,C,=,p,L,=,p,S,=,p,A,=.25,H,a,:The population proportions are,not,p,C,=.25,p,L,=.25,p,S,=.25,and,p,A,=.25,HypothesesMultinomial Distribu,Rejection Rule,2,7.815,Do Not Reject,H,0,Reject,H,0,Multinomial Distribution Goodness of Fit Test,With,=.05 and,k,-1=4-1=3,degrees of freedom,Reject,H,0,if,p,-value,7.815.,Rejection Rule2 7.815Do Not,Expected Frequencies,Test Statistic,Multinomial Distribution Goodness of Fit Test,e,1,=.25(100)=25,e,2,=.25(100)=25,e,3,=.25(100)=25,e,4,=.25(100)=25,=1+1+4+4,=10,Expected FrequenciesMultinomia,Multinomial Distribution Goodness of Fit Test,Conclusion Using the,p,-Value Approach,The,p,-value,7.815,Conclusion Using the Critical,Test of Independence:Contingency Tables,1.,Set up the null and alternative hypotheses.,2.,Select a random sample and record the observed,frequency,f,ij,for each cell of the contingency table.,3.,Compute the expected frequency,e,ij,for each cell.,H,0,:The column variable is independent of,the row variable,H,a,:The column variable is,not,independent,of the row variable,Test of Independence:Conting,Test of Independence:Contingency Tables,5.,Determine the rejection rule.,Reject,H,0,if,p,-value,$99,000 12 14 16 3,$99,000 18 6 19 12,Contingency Table(Independence)Test,Example:Finger Lakes Homes(B),Price Colonial L,Hypotheses,Contingency Table(Independence)Test,H,0,:Price of the home,is,independent of the,style of the home that is purchased,H,a,:Price of the home,is not,independent of the,style of the home that is purchased,HypothesesContingency Table(I,Expected Frequencies,Contingency Table(Independence)Test,Price,Colonial Log Split-Level A-Frame Total,$99K,Total,30 20 35 15 100,12 14 16 3 45,18 6 19 12 55,Expected FrequenciesContingenc,Rejection Rule,Contingency Table(Independence)Test,With,=.05 and(2-1)(4-1)=3 d.f.,Reject,H,0,if,p,-value,7.815,=.1364+2.2727+.+2.0833=9.149,Test Statistic,Rejection RuleContingency Tabl,Conclusion Using the,p,-Value Approach,The,p,-value,7.815,Conclusion Using the Critical,Goodness of Fit Test:Poisson Distribution,1.,State the null and alternative hypotheses.,2.,Select a random sample and,a.,Record the obs
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