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### Chap08

1. 1. Chap 8-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.Chapter 8Estimation: Single PopulationStatistics forBusiness and Economics6thEdition
2. 2. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-2Chapter GoalsAfter completing this chapter, you should beable to: Distinguish between a point estimate and aconfidence interval estimate Construct and interpret a confidence intervalestimate for a single population mean using boththe Z and t distributions Form and interpret a confidence interval estimatefor a single population proportion
3. 3. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-3Confidence IntervalsContent of this chapter Confidence Intervals for the PopulationMean, μ when Population Variance σ2is Known when Population Variance σ2is Unknown Confidence Intervals for the PopulationProportion, (large samples)pˆ
4. 4. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-4Definitions An estimator of a population parameter is a random variable that depends on sampleinformation . . . whose value provides an approximation to thisunknown parameter A specific value of that random variable iscalled an estimate
5. 5. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-5Point and Interval Estimates A point estimate is a single number, a confidence interval provides additionalinformation about variabilityPoint EstimateLowerConfidenceLimitUpperConfidenceLimitWidth ofconfidence interval
6. 6. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-6We can estimate aPopulation Parameter …Point Estimateswith a SampleStatistic(a Point Estimate)MeanProportion Pxμpˆ
7. 7. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-7Unbiasedness A point estimator is said to be anunbiased estimator of the parameter θ if theexpected value, or mean, of the samplingdistribution of is θ, Examples: The sample mean is an unbiased estimator of μ The sample variance is an unbiased estimator of σ2 The sample proportion is an unbiased estimator of Pθˆθˆθ)θE( =ˆ
8. 8. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-8 is an unbiased estimator, is biased:1θˆ2θˆθˆθ1θˆ2θˆUnbiasedness(continued)
9. 9. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-9Bias Let be an estimator of θ The bias in is defined as the differencebetween its mean and θ The bias of an unbiased estimator is 0θˆθˆθ)θE()θBias( −= ˆˆ
10. 10. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-10Consistency Let be an estimator of θ is a consistent estimator of θ if thedifference between the expected value of andθ decreases as the sample size increases Consistency is desired when unbiasedestimators cannot be obtainedθˆθˆθˆ
11. 11. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-11Most Efficient Estimator Suppose there are several unbiased estimators of θ The most efficient estimator or the minimum varianceunbiased estimator of θ is the unbiased estimator with thesmallest variance Let and be two unbiased estimators of θ, based onthe same number of sample observations. Then,is said to be more efficient than if The relative efficiency of with respect to is the ratioof their variances:)θVar()θVar( 21ˆˆ <)θVar()θVar(EfficiencyRelative12ˆˆ=1θˆ2θˆ1θˆ2θˆ1θˆ2θˆ
12. 12. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-12Confidence Intervals How much uncertainty is associated with apoint estimate of a population parameter? An interval estimate provides moreinformation about a population characteristicthan does a point estimate Such interval estimates are called confidenceintervals
13. 13. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-13Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in samplestatistics from sample to sample Based on observation from 1 sample Gives information about closeness tounknown population parameters Stated in terms of level of confidenceCan never be 100% confident
14. 14. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-14Confidence Interval andConfidence Level If P(a < θ < b) = 1 - α then the interval from ato b is called a 100(1 - α)% confidenceinterval of θ. The quantity (1 - α) is called the confidencelevel of the interval (α between 0 and 1) In repeated samples of the population, the true valueof the parameter θ would be contained in 100(1 - α)% of intervals calculated this way. The confidence interval calculated in this manner iswritten as a < θ < b with 100(1 - α)% confidence
15. 15. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-15Estimation Process(mean, μ, isunknown)PopulationRandom SampleMeanX = 50SampleI am 95%confident thatμ is between40 & 60.
16. 16. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-16Confidence Level, (1-α) Suppose confidence level = 95% Also written (1 - α) = 0.95 A relative frequency interpretation: From repeated samples, 95% of all theconfidence intervals that can be constructed willcontain the unknown true parameter A specific interval either will contain or willnot contain the true parameter No probability involved in a specific interval(continued)
17. 17. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-17General Formula The general formula for all confidenceintervals is: The value of the reliability factordepends on the desired level ofconfidencePoint Estimate ± (Reliability Factor)(Standard Error)
18. 18. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-18Confidence IntervalsPopulationMeanσ2UnknownConfidenceIntervalsPopulationProportionσ2Known
19. 19. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-19Confidence Interval for μ(σ2Known) Assumptions Population variance σ2is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate:(where zα/2 is the normal distribution value for a probability of α/2 ineach tail)nσzxμnσzx α/2α/2 +<<−
20. 20. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-20Margin of Error The confidence interval, Can also be written aswhere ME is called the margin of error The interval width, w, is equal to twice the margin oferrornσzxμnσzx α/2α/2 +<<−MEx ±nσzME α/2=
21. 21. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-21Reducing the Margin of ErrorThe margin of error can be reduced if the population standard deviation can be reduced (σ↓) The sample size is increased (n↑) The confidence level is decreased, (1 – α) ↓nσzME α/2=
22. 22. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-22Finding the Reliability Factor, zα/2 Consider a 95% confidence interval:z = -1.96 z = 1.96.951 =α−.0252α= .0252α=Point EstimateLowerConfidenceLimitUpperConfidenceLimitZ units:X units: Point Estimate0 Find z.025 = ±1.96 from the standard normal distribution table
23. 23. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-23Common Levels of Confidence Commonly used confidence levels are 90%,95%, and 99%ConfidenceLevelConfidenceCoefficient, Zα/2 value1.281.6451.962.332.583.083.27.80.90.95.98.99.998.99980%90%95%98%99%99.8%99.9%α−1
24. 24. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-24μμx=Intervals and Level of ConfidenceConfidence IntervalsIntervalsextend fromto100(1-α)%of intervalsconstructedcontain μ;100(α)% donot.Sampling Distribution of the Meannσzx −nσzx +xx1x2/2α /2αα−1
25. 25. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-25Example A sample of 11 circuits from a large normalpopulation has a mean resistance of 2.20ohms. We know from past testing that thepopulation standard deviation is 0.35 ohms. Determine a 95% confidence interval for thetrue mean resistance of the population.
26. 26. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-262.4068μ1.9932.20682.20)11(.35/1.962.20nσzx<<±=±=±Example A sample of 11 circuits from a large normalpopulation has a mean resistance of 2.20ohms. We know from past testing that thepopulation standard deviation is .35 ohms. Solution:(continued)
27. 27. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-27Interpretation We are 95% confident that the true meanresistance is between 1.9932 and 2.4068ohms Although the true mean may or may not bein this interval, 95% of intervals formed inthis manner will contain the true mean
28. 28. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-28Confidence IntervalsPopulationMeanConfidenceIntervalsPopulationProportionσ2Unknownσ2Known
29. 29. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-29Student’s t Distribution Consider a random sample of n observations with mean x and standard deviation s from a normally distributed population with mean μ Then the variablefollows the Student’s t distribution with (n - 1) degreesof freedomns/μxt−=
30. 30. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-30 If the population standard deviation σ isunknown, we can substitute the samplestandard deviation, s This introduces extra uncertainty, sinces is variable from sample to sample So we use the t distribution instead ofthe normal distributionConfidence Interval for μ(σ2Unknown)
31. 31. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-31 Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student’s t Distribution Confidence Interval Estimate:where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. andan area of α/2 in each tail:Confidence Interval for μ(σ Unknown)nStxμnStx α/21,-nα/21,-n +<<−(continued)α/2)tP(t α/21,n1n => −−
32. 32. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-32Student’s t Distribution The t is a family of distributions The t value depends on degrees offreedom (d.f.) Number of observations that are free to vary aftersample mean has been calculatedd.f. = n - 1
33. 33. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-33Student’s t Distributiont0t (df = 5)t (df = 13)t-distributions are bell-shaped and symmetric, buthave ‘fatter’ tails than thenormalStandardNormal(t with df = ∞)Note: t Z as n increases
34. 34. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-34Student’s t TableUpper Tail Areadf .10 .025.051 12.70623 3.182t0 2.920The body of the tablecontains t values, notprobabilitiesLet: n = 3df = n - 1 = 2α = .10α/2 =.05α/2 = .053.0781.8861.6386.3142.9202.3534.303
35. 35. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-35t distribution valuesWith comparison to the Z valueConfidence t t t ZLevel (10 d.f.) (20 d.f.) (30 d.f.) ____.80 1.372 1.325 1.310 1.282.90 1.812 1.725 1.697 1.645.95 2.228 2.086 2.042 1.960.99 3.169 2.845 2.750 2.576Note: t Z as n increases
36. 36. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-36ExampleA random sample of n = 25 has x = 50 ands = 8. Form a 95% confidence interval for μ d.f. = n – 1 = 24, soThe confidence interval is2.0639tt 24,.025α/21,n ==−53.302μ46.698258(2.0639)50μ258(2.0639)50nStxμnStx α/21,-nα/21,-n<<+<<−+<<−
37. 37. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-37Confidence IntervalsPopulationMeanσ UnknownConfidenceIntervalsPopulationProportionσ Known
38. 38. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-38Confidence Intervals for thePopulation Proportion, p An interval estimate for the populationproportion ( P ) can be calculated byadding an allowance for uncertainty tothe sample proportion ( )pˆ
39. 39. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-39Confidence Intervals for thePopulation Proportion, p Recall that the distribution of the sampleproportion is approximately normal if thesample size is large, with standard deviation We will estimate this with sample data:(continued)n)p(1p ˆˆ −nP)P(1σP−=
40. 40. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-40Confidence Interval Endpoints Upper and lower confidence limits for thepopulation proportion are calculated with theformula where zα/2 is the standard normal value for the level of confidence desired is the sample proportion n is the sample sizen)p(1pzpPn)p(1pzp α/2α/2ˆˆˆˆˆˆ −+<<−−pˆ
41. 41. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-41Example A random sample of 100 peopleshows that 25 are left-handed. Form a 95% confidence interval forthe true proportion of left-handers
42. 42. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-42Example A random sample of 100 people showsthat 25 are left-handed. Form a 95%confidence interval for the true proportionof left-handers.(continued)0.3349P0.1651100.25(.75)1.9610025P100.25(.75)1.9610025n)p(1pzpPn)p(1pzp α/2α/2<<+<<−−+<<−−ˆˆˆˆˆˆ
43. 43. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-43Interpretation We are 95% confident that the truepercentage of left-handers in the populationis between16.51% and 33.49%. Although the interval from 0.1651 to 0.3349may or may not contain the true proportion,95% of intervals formed from samples ofsize 100 in this manner will contain the trueproportion.
44. 44. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-44PHStat Interval Optionsoptions
45. 45. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-45Using PHStat(for μ, σ unknown)A random sample of n = 25 has X = 50 andS = 8. Form a 95% confidence interval for μ
46. 46. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 8-46Chapter Summary Introduced the concept of confidenceintervals Discussed point estimates Developed confidence interval estimates Created confidence interval estimates for themean (σ2known) Introduced the Student’s t distribution Determined confidence interval estimates forthe mean (σ2unknown) Created confidence interval estimates for theproportion