Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-1Chapter 8Confidence Interval EstimationBusiness Statistics:A First Course5th EditionChapter ProblemSaxon Home ImprovementSaxon Home ImprovementBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-4Learning ObjectivesIn this chapter, you learn: nTo construct and interpret confidence interval estimates for the mean and the proportionnHow to determine the sample size necessary to develop a confidence interval for the mean or proportionBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-5Chapter OutlineContent of this chapternConfidence Intervals for the Population Mean, nwhen Population Standard Deviation is Knownnwhen Population Standard Deviation is UnknownnConfidence Intervals for the Population Proportion, nDetermining the Required Sample SizeBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-6Point and Interval EstimatesnA point estimate is a single number na confidence interval provides additional information about the variability of the estimatePoint EstimateLower Confidence LimitUpperConfidence LimitWidth of confidence intervalBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-7We can estimate a Population Parameter Point Estimateswith a SampleStatistic(a Point Estimate)MeanProportionpXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-8Confidence IntervalsnHow much uncertainty is associated with a point estimate of a population parameter?nAn interval estimate provides more information about a population characteristic than does a point estimatenSuch interval estimates are called confidence intervalsBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-9Confidence Interval EstimatenAn interval gives a range of values:nTakes into consideration variation in sample statistics from sample to samplenBased on observations from 1 samplenGives information about closeness to unknown population parametersnStated in terms of level of confidencene.g. 95% confident, 99% confidentnCan never be 100% confidentBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-10Confidence Interval ExampleCereal fill examplen Population has = 368 and = 15.n If you take a sample of size n = 25 you known368 1.96 * 15 / = (362.12, 373.88) contains 95% of the sample meansnWhen you dont know , you use X to estimate nIf X = 362.3 the interval is 362.3 1.96 * 15 / = (356.42, 368.18)nSince 356.42 368.18, the interval based on this sample makes a correct statement about .But what about the intervals from other possible samples of size 25?Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-11Confidence Interval Example(continued)Sample #XLowerLimitUpperLimitContain?1362.30356.42368.18Yes2369.50363.62375.38Yes3360.00354.12365.88No4362.12356.24368.00Yes5373.88368.00379.76YesPoint and Interval EstimatesBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-13Confidence Interval ExamplenIn practice you only take one sample of size nnIn practice you do not know so you do not know if the interval actually contains nHowever you do know that 95% of the intervals formed in this manner will contain nThus, based on the one sample, you actually selected you can be 95% confident your interval will contain (this is a 95% confidence interval)(continued)Note: 95% confidence is based on the fact that we used Z = 1.96.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-14Estimation Process(mean, , is unknown)PopulationRandom SampleMean X = 50SampleI am 95% confident that is between 40 & 60.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-15General FormulanThe general formula for all confidence intervals is:Point Estimate (Critical Value)(Standard Error)Where:Point Estimate is the sample statistic estimating the population parameter of interestCritical Value is a table value based on the sampling distribution of the point estimate and the desired confidence levelStandard Error is the standard deviation of the point estimateBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-16Confidence LevelnConfidence LevelnThe confidence that the interval will contain the unknown population parameternA percentage (less than 100%)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-17Confidence Level, (1-)nSuppose confidence level = 95% nAlso written (1 - ) = 0.95, (so = 0.05)nA relative frequency interpretation:n95% of all the confidence intervals that can be constructed will contain the unknown true parameternA specific interval either will contain or will not contain the true parameternNo probability involved in a specific interval(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-18Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion KnownBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-19Confidence Interval for ( Known) nAssumptionsnPopulation standard deviation is knownnPopulation is normally distributednIf population is not normal, use large samplenConfidence interval estimate:n where is the point estimate Z/2 is the normal distribution critical value for a probability of /2 in each tail is the standard error Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-20Finding the Critical Value, Z/2nConsider a 95% confidence interval:Z/2 = -1.96Z/2 = 1.96Point EstimateLower Confidence LimitUpperConfidence LimitZ units:X units:Point Estimate0Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-21Common Levels of ConfidencenCommonly used confidence levels are 90%, 95%, and 99%Confidence LevelConfidence Coefficient, Z/2 value1.281.6451.962.332.583.083.270.800.900.950.980.990.9980.99980%90%95%98%99%99.8%99.9%Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-22Intervals and Level of ConfidenceConfidence Intervals Intervals extend from to (1-)x100%of intervals constructed contain ; ()x100% do not.Sampling Distribution of the Meanxx1x2Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-23ExamplenA sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. nDetermine a 95% confidence interval for the true mean resistance of the population.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-24ExamplenA sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. nSolution:(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-25InterpretationnWe are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms nAlthough the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true meanBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-26Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion KnownBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-27Do You Ever Truly Know ?nProbably not!nIn virtually all real world business situations, is not known.nIf there is a situation where is known then is also known (since to calculate you need to know .)nIf you truly know there would be no need to gather a sample to estimate it.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-28nIf the population standard deviation is unknown, we can substitute the sample standard deviation, S nThis introduces extra uncertainty, since S is variable from sample to samplenSo we use the t distribution instead of the normal distributionConfidence Interval for ( Unknown) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-29nAssumptionsnPopulation standard deviation is unknownnPopulation is normally distributednIf population is not normal, use large samplenUse Students t DistributionnConfidence Interval Estimate:n(where t/2 is the critical value of the t distribution with n -1 degrees of freedom and an area of /2 in each tail) Confidence Interval for ( Unknown) (continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-30Students t DistributionnThe t is a family of distributionsnThe t/2 value depends on degrees of freedom (d.f.)nNumber of observations that are free to vary after sample mean has been calculatedd.f. = n - 1Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-31If the mean of these three values is 8.0, then X3 must be 9 (i.e., X3 is not free to vary)Degrees of Freedom (df)Here, n = 3, so degrees of freedom = n 1 = 3 1 = 2(2 values can be any numbers, but the third is not free to vary for a given mean)Idea: Number of observations that are free to vary after sample mean has been calculatedExample: Suppose the mean of 3 numbers is 8.0 Let X1 = 7Let X2 = 8What is X3?Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-32Students t Distributiont0t (df = 5) t (df = 13)t-distributions are bell-shaped and symmetric, but have fatter tails than the normalStandard Normal(t with df = )Note: t Z as n increasesBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-33Students t TableUpper Tail Areadf.25.10.0511.000 3.078 6.31420.817 1.886 2.92030.765 1.638 2.353t02.920The body of the table contains t values, not probabilitiesLet: n = 3 df = n - 1 = 2 = 0.10 /2 = 0.05/2 = 0.05Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-34Selected t distribution valuesWith comparison to the Z valueConfidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ( d.f.) 0.80 1.372 1.325 1.310 1.28 0.90 1.812 1.725 1.697 1.645 0.95 2.228 2.086 2.042 1.96 0.99 3.169 2.845 2.750 2.58Note: t Z as n increasesBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-35Example of t distribution confidence interval A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for nd.f. = n 1 = 24, soThe confidence interval is 46.698 53.302Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-36Example of t distribution confidence intervalnInterpreting this interval requires the assumption that the population you are sampling from is approximately a normal distribution (especially since n is only 25).nThis condition can be checked by creating a:nNormal probability plot ornBoxplot(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-37Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion KnownBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-38Confidence Intervals for the Population Proportion, nAn interval estimate for the population proportion ( ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-39Confidence Intervals for the Population Proportion, nRecall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviationnWe will estimate this with sample data:(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-40Confidence Interval EndpointsnUpper and lower confidence limits for the population proportion are calculated with the formulanwhere nZ/2 is the standard normal value for the level of confidence desirednp is the sample proportionnn is the sample sizenNote: must have np 5 and n(1-p) 5Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-41ExamplenA random sample of 100 people shows that 25 are left-handed. nForm a 95% confidence interval for the true proportion of left-handersBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-42ExamplenA random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-43InterpretationnWe are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. nAlthough the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-44Determining Sample SizeFor the MeanDeterminingSample SizeFor theProportionBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-45Sampling ErrornThe required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - )nThe margin of error is also called sampling errornthe amount of imprecision in the estimate of the population parameternthe amount added and subtracted to the point estimate to form the confidence interval Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-46Determining Sample SizeFor the MeanDeterminingSample SizeSampling error (margin of error)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-47Determining Sample SizeFor the MeanDeterminingSample Size(continued)Now solve for n to getBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-48Determining Sample SizenTo determine the required sample size for the mean, you must know:nThe desired level of confidence (1 - ), which determines the critical value, Z/2nThe acceptable sampling error, enThe standard deviation, (continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-49Required Sample Size ExampleIf = 45, what sample size is needed to estimate the mean within 5 with 90% confidence? (Always round up)So the required sample size is n = 220Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-50If is unknownnIf unknown, can be estimated when using the required sample size formulanUse a value for that is expected to be at least as large as the true nSelect a pilot sample and estimate with the sample standard deviation, S Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-51Determining Sample SizeDeterminingSample SizeFor theProportionNow solve for n to get(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-52Determining Sample SizenTo determine the required sample size for the proportion, you must know:nThe desired level of confidence (1 - ), which determines the critical value, Z/2nThe acceptable sampling error, enThe true proportion of events of interest, n can be estimated with a pilot sample if necessary (or conservatively use 0.5 as an estimate of )(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-53Required Sample Size ExampleHow large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p = 0.12)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-54Required Sample Size ExampleSolution:For 95% confidence, use Z/2 = 1.96e = 0.03p = 0.12, so use this to estimate So use n = 451(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-55Ethical IssuesnA confidence interval estimate (reflecting sampling error) should always be included when reporting a point estimate nThe level of confidence should always be reported nThe sample size should be reportednAn interpretation of the confidence interval estimate should also be providedBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-56Chapter SummarynIntroduced the concept of confidence intervalsnDiscussed point estimatesnDeveloped confidence interval estimatesnCreated confidence interval estimates for the mean ( known)nDetermined confidence interval estimates for the mean ( unknown)nCreated confidence interval estimates for the proportion nDetermined required sample size for mean and proportion settingsnAddressed confidence interval estimation and ethical issues