Chap09

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Chap09

  1. 1. Chap 9-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.Chapter 9Estimation: Additional TopicsStatistics forBusiness and Economics6thEdition
  2. 2. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-2Chapter GoalsAfter completing this chapter, you should be able to: Form confidence intervals for the mean difference from dependentsamples Form confidence intervals for the difference between twoindependent population means (standard deviations known orunknown) Compute confidence interval limits for the difference between twoindependent population proportions Create confidence intervals for a population variance Find chi-square values from the chi-square distribution table Determine the required sample size to estimate a mean orproportion within a specified margin of error
  3. 3. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-3Estimation: Additional TopicsChapter TopicsPopulationMeans,IndependentSamplesPopulationMeans,DependentSamplesPopulationVarianceGroup 1 vs.independentGroup 2Same groupbefore vs. aftertreatmentVariance of anormal distributionExamples:PopulationProportionsProportion 1 vs.Proportion 2
  4. 4. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-4Dependent SamplesTests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally DistributedDependentsamplesdi = xi - yi
  5. 5. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-5Mean DifferenceThe ithpaired difference is di , wheredi = xi - yiThe point estimate forthe population meanpaired difference is d :nddn1ii∑==n is the number of matched pairs in the sample1n)d(dSn1i2id−−=∑=The samplestandarddeviation is:Dependentsamples
  6. 6. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-6Confidence Interval forMean DifferenceThe confidence interval for differencebetween population means, μd , isWheren = the sample size(number of matched pairs in the paired sample)nStdμnStd dα/21,nddα/21,n −− +<<−Dependentsamples
  7. 7. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-7 The margin of error is tn-1,α/2 is the value from the Student’s tdistribution with (n – 1) degrees of freedomfor whichConfidence Interval forMean Difference(continued)2α)tP(t α/21,n1n => −−nstME dα/21,n−=Dependentsamples
  8. 8. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-8 Six people sign up for a weight loss program. Youcollect the following data:Paired Samples ExampleWeight:Person Before (x) After (y) Difference, di1 136 125 112 205 195 103 157 150 74 138 140 - 25 175 165 106 166 160 642d =Σ din4.821n)d(dS2id=−−=∑= 7.0
  9. 9. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-9 For a 95% confidence level, the appropriate t value is tn-1,α/2 = t5,.025 = 2.571 The 95% confidence interval for the difference betweenmeans, μd , is12.06μ1.9464.82(2.571)7μ64.82(2.571)7nStdμnStddddα/21,nddα/21,n<<−+<<−+<<− −−Paired Samples Example(continued)Since this interval contains zero, we cannot be 95% confident, given thislimited data, that the weight loss program helps people lose weight
  10. 10. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-10Difference Between Two MeansPopulation means,independentsamplesGoal: Form a confidence intervalfor the difference between twopopulation means, μx – μyx – y Different data sources Unrelated IndependentSample selected from one population has no effect on thesample selected from the other population The point estimate is the difference between the twosample means:
  11. 11. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-11Difference Between Two MeansPopulation means,independentsamplesConfidence interval uses zα/2Confidence interval uses a valuefrom the Student’s t distributionσx2and σy2assumed equalσx2and σy2knownσx2and σy2unknownσx2and σy2assumed unequal(continued)
  12. 12. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-12Population means,independentsamplesσx2and σy2KnownAssumptions: Samples are randomly andindependently drawn both population distributionsare normal Population variances areknown*σx2and σy2knownσx2and σy2unknown
  13. 13. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-13Population means,independentsamples…and the random variablehas a standard normal distributionWhen σx and σy are known andboth populations are normal, thevariance of X – Y isy2yx2x2YXnσnσσ +=−(continued)*Y2yX2xYXnσnσ)μ(μ)yx(Z+−−−=σx2and σy2knownσx2and σy2unknownσx2and σy2Known
  14. 14. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-14Population means,independentsamplesThe confidence interval forμx – μy is:Confidence Interval,σx2and σy2Known*y2Yx2Xα/2YXy2Yx2Xα/2nσnσz)yx(μμnσnσz)yx( ++−<−<+−−σx2and σy2knownσx2and σy2unknown
  15. 15. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-15Population means,independentsamplesσx2and σy2Unknown,Assumed EqualAssumptions: Samples are randomly andindependently drawn Populations are normallydistributed Population variances areunknown but assumed equal*σx2and σy2assumed equalσx2and σy2knownσx2and σy2unknownσx2and σy2assumed unequal
  16. 16. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-16Population means,independentsamples(continued)Forming intervalestimates: The population variancesare assumed equal, so usethe two sample standarddeviations and pool them toestimate σ use a t value with(nx + ny – 2) degrees offreedom*σx2and σy2assumed equalσx2and σy2knownσx2and σy2unknownσx2and σy2assumed unequalσx2 and σy2Unknown,Assumed Equal
  17. 17. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-17Population means,independentsamplesThe pooled variance is(continued)*2nn1)s(n1)s(nsyx2yy2xx2p−+−+−=σx2and σy2assumed equalσx2and σy2knownσx2and σy2unknownσx2and σy2assumed unequalσx2and σy2Unknown,Assumed Equal
  18. 18. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-18The confidence interval forμ1 – μ2 is:Where*Confidence Interval,σx2and σy2Unknown, Equalσx2and σy2assumed equalσx2and σy2unknownσx2and σy2assumed unequaly2px2pα/22,nnYXy2px2pα/22,nnnsnst)yx(μμnsnst)yx( yxyx++−<−<+−− −+−+2nn1)s(n1)s(nsyx2yy2xx2p−+−+−=
  19. 19. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-19Pooled Variance ExampleYou are testing two computer processors for speed.Form a confidence interval for the difference in CPUspeed. You collect the following speed data (in Mhz):CPUx CPUyNumber Tested 17 14Sample mean 3004 2538Sample std dev 74 56Assume both populations arenormal with equal variances,and use 95% confidence
  20. 20. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-20Calculating the Pooled Variance( ) ( ) ( ) ( ) 4427.031)141)-(1756114741171)n(nS1nS1nS22y2yy2xx2p =−+−+−=−+−−+−=(()1xThe pooled variance is:The t value for a 95% confidence interval is:2.045tt 0.025,29α/2,2nn yx==−+
  21. 21. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-21Calculating the Confidence Limits The 95% confidence interval isy2px2pα/22,nnYXy2px2pα/22,nnnsnst)yx(μμnsnst)yx( yxyx++−<−<+−− −+−+144427.03174427.03(2.054)2538)(3004μμ144427.03174427.03(2.054)2538)(3004 YX ++−<−<+−−515.31μμ416.69 YX <−<We are 95% confident that the mean difference inCPU speed is between 416.69 and 515.31 Mhz.
  22. 22. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-22Population means,independentsamplesσx2and σy2Unknown,Assumed UnequalAssumptions: Samples are randomly andindependently drawn Populations are normallydistributed Population variances areunknown and assumedunequal*σx2and σy2assumed equalσx2and σy2knownσx2and σy2unknownσx2and σy2assumed unequal
  23. 23. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-23Population means,independentsamplesσx2and σy2Unknown,Assumed Unequal(continued)Forming interval estimates: The population variances areassumed unequal, so a pooledvariance is not appropriate use a t value with ν degreesof freedom, whereσx2and σy2knownσx2and σy2unknown*σx2and σy2assumed equalσx2and σy2assumed unequal1)/(nns1)/(nns)ns()ns(y2y2yx2x2x2y2yx2x−+−+=v
  24. 24. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-24The confidence interval forμ1 – μ2 is:*Confidence Interval,σx2and σy2Unknown, Unequalσx2and σy2assumed equalσx2and σy2unknownσx2and σy2assumed unequaly2yx2xα/2,YXy2yx2xα/2,nsnst)yx(μμnsnst)yx( ++−<−<+−− νν1)/(nns1)/(nns)ns()ns(y2y2yx2x2x2y2yx2x−+−+=vWhere
  25. 25. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-25Two Population ProportionsGoal: Form a confidence interval forthe difference between twopopulation proportions, Px – PyThe point estimate forthe difference isPopulationproportionsAssumptions:Both sample sizes are large (generally atleast 40 observations in each sample)yx pp ˆˆ −
  26. 26. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-26Two Population ProportionsPopulationproportions(continued) The random variableis approximately normally distributedyyyxxxyxyxn)p(1pn)p(1p)p(p)pp(Zˆˆˆˆˆˆ−+−−−−=
  27. 27. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-27Confidence Interval forTwo Population ProportionsPopulationproportionsThe confidence limits forPx – Py are:yyyxxxyxn)p(1pn)p(1pZ)pp(ˆˆˆˆˆˆ 2/−+−±− α
  28. 28. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-28Example:Two Population ProportionsForm a 90% confidence interval for thedifference between the proportion ofmen and the proportion of women whohave college degrees. In a random sample, 26 of 50 men and28 of 40 women had an earned collegedegree
  29. 29. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-29Example:Two Population ProportionsMen:Women:0.1012400.70(0.30)500.52(0.48)n)p(1pn)p(1pyyyxxx=+=−+− ˆˆˆˆ0.525026px ==ˆ0.704028py ==ˆ(continued)For 90% confidence, Zα/2 = 1.645
  30. 30. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-30Example:Two Population ProportionsThe confidence limits are:so the confidence interval is-0.3465 < Px – Py < -0.0135Since this interval does not contain zero we are 90% confident that thetwo proportions are not equal(continued)(0.1012)1.645.70)(.52n)p(1pn)p(1pZ)pp(yyyxxxα/2yx±−=−+−±−ˆˆˆˆˆˆ
  31. 31. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-31Confidence Intervals for thePopulation VariancePopulationVariance Goal: Form a confidence intervalfor the population variance, σ2 The confidence interval is based onthe sample variance, s2 Assumed: the population isnormally distributed
  32. 32. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-32Confidence Intervals for thePopulation VariancePopulationVarianceThe random variable2221nσ1)s(n−=−χfollows a chi-square distributionwith (n – 1) degrees of freedom(continued)The chi-square value denotes the number for which2,1n αχ −α)P( 2α,1n21n => −− χχ
  33. 33. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-33Confidence Intervals for thePopulation VariancePopulationVarianceThe (1 - α)% confidence intervalfor the population variance is2/2-1,1n222/2,1n21)s(nσ1)s(nαα χχ −−−<<−(continued)
  34. 34. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-34ExampleYou are testing the speed of a computer processor.You collect the following data (in Mhz):CPUxSample size 17Sample mean 3004Sample std dev 74Assume the population is normal.Determine the 95% confidence interval for σx2
  35. 35. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-35Finding the Chi-square Values n = 17 so the chi-square distribution has (n – 1) = 16degrees of freedom α = 0.05, so use the the chi-square values with area0.025 in each tail:probabilityα/2 = .025χ216χ216= 28.856.9128.8520.975,162/2-1,1n20.025,162/2,1n====−−χχχχααχ216 = 6.91probabilityα/2 = .025
  36. 36. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-36Calculating the Confidence Limits The 95% confidence interval isConverting to standard deviation, we are 95%confident that the population standard deviation ofCPU speed is between 55.1 and 112.6 Mhz2/2-1,1n222/2,1n21)s(nσ1)s(nαα χχ −−−<<−6.911)(74)(17σ28.851)(74)(17 222−<<−12683σ3037 2<<
  37. 37. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-37Sample PHStat Output
  38. 38. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-38Sample PHStat OutputInputOutput(continued)
  39. 39. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-39Sample Size DeterminationFor theMeanDeterminingSample SizeFor theProportion
  40. 40. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-40Margin of Error The required sample size can be found to reacha desired margin of error (ME) with a specifiedlevel of confidence (1 - α) The margin of error is also called sampling error the amount of imprecision in the estimate of thepopulation parameter the amount added and subtracted to the pointestimate to form the confidence interval
  41. 41. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-41For theMeanDeterminingSample Sizenσzx α/2±nσzME α/2=Margin of Error(sampling error)Sample Size Determination
  42. 42. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-42For theMeanDeterminingSample SizenσzME α/2=(continued)222α/2MEσzn =Now solvefor n to getSample Size Determination
  43. 43. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-43 To determine the required sample size for themean, you must know: The desired level of confidence (1 - α), whichdetermines the zα/2 value The acceptable margin of error (sampling error), ME The standard deviation, σ(continued)Sample Size Determination
  44. 44. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-44Required Sample Size ExampleIf σ = 45, what sample size is needed toestimate the mean within ± 5 with 90%confidence?(Always round up)219.195(45)(1.645)MEσzn 222222α/2===So the required sample size is n = 220
  45. 45. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-45n)p(1pzp α/2ˆˆˆ −±n)p(1pzME α/2ˆˆ −=DeterminingSample SizeFor theProportionMargin of Error(sampling error)Sample Size Determination
  46. 46. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-46DeterminingSample SizeFor theProportion22α/2MEz0.25n =Substitute0.25 forand solve forn to get(continued)Sample Size Determinationn)p(1pzME α/2ˆˆ −=cannotbe larger than0.25, when =0.5)p(1p ˆˆ −pˆ)p(1p ˆˆ −
  47. 47. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-47 The sample and population proportions, and P, aregenerally not known (since no sample has been takenyet) P(1 – P) = 0.25 generates the largest possible marginof error (so guarantees that the resulting sample sizewill meet the desired level of confidence) To determine the required sample size for theproportion, you must know: The desired level of confidence (1 - α), which determines thecritical zα/2 value The acceptable sampling error (margin of error), ME Estimate P(1 – P) = 0.25(continued)Sample Size Determinationpˆ
  48. 48. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-48Required Sample Size ExampleHow large a sample would be necessaryto estimate the true proportion defective ina large population within ±3%, with 95%confidence?
  49. 49. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-49Required Sample Size ExampleSolution:For 95% confidence, use z0.025 = 1.96ME = 0.03Estimate P(1 – P) = 0.25So use n = 1068(continued)1067.11(0.03)6)(0.25)(1.9MEz0.25n 2222α/2===
  50. 50. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-50PHStat Sample Size Options
  51. 51. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 9-51Chapter Summary Compared two dependent samples (paired samples) Formed confidence intervals for the paired difference Compared two independent samples Formed confidence intervals for the difference between twomeans, population variance known, using z Formed confidence intervals for the differences between twomeans, population variance unknown, using t Formed confidence intervals for the differences between twopopulation proportions Formed confidence intervals for the population varianceusing the chi-square distribution Determined required sample size to meet confidenceand margin of error requirements

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