應(yīng)用統(tǒng)計學(xué)英文課件BusinessStatisticsCh08ConfidenceIntervalEstima

1、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 chap

2、ter, you learn: To construct and interpret confidence interval estimates for the mean and the proportionHow 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

3、chapterConfidence Intervals for the Population Mean, when Population Standard Deviation is Knownwhen Population Standard Deviation is UnknownConfidence Intervals for the Population Proportion, Determining the Required Sample SizeBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-6Point an

4、d Interval EstimatesA point estimate is a single number a 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 ca

5、n estimate a Population Parameter Point Estimateswith a SampleStatistic(a Point Estimate)MeanProportionpXBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-8Confidence IntervalsHow much uncertainty is associated with a point estimate of a population parameter?An interval estimate provides

6、 more information about a population characteristic than does a point estimateSuch interval estimates are called confidence intervalsBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-9Confidence Interval EstimateAn interval gives a range of values:Takes into consideration variation in sa

7、mple statistics from sample to sampleBased on observations from 1 sampleGives information about closeness to unknown population parametersStated in terms of level of confidencee.g. 95% confident, 99% confidentCan never be 100% confidentBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-10

8、Confidence Interval ExampleCereal fill example Population has = 368 and = 15. If you take a sample of size n = 25 you know368 1.96 * 15 / = (362.12, 373.88) contains 95% of the sample meansWhen you dont know , you use X to estimate If X = 362.3 the interval is 362.3 1.96 * 15 / = (356.42, 368.18)Sin

9、ce 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?1362.303

10、56.42368.18Yes2369.50363.62375.38Yes3360.00354.12365.88No4362.12356.24368.00Yes5373.88368.00379.76YesPoint and Interval EstimatesBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-13Confidence Interval ExampleIn practice you only take one sample of size nIn practice you do not know so you

11、 do not know if the interval actually contains However you do know that 95% of the intervals formed in this manner will contain Thus, 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

12、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

13、FormulaThe 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 desir

14、ed confidence levelStandard Error is the standard deviation of the point estimateBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-16Confidence LevelConfidence LevelThe confidence that the interval will contain the unknown population parameterA percentage (less than 100%)Basic Business S

15、tatistics, 11e 2009 Prentice-Hall, Inc.Chap 8-17Confidence Level, (1-)Suppose confidence level = 95% Also written (1 - ) = 0.95, (so = 0.05)A relative frequency interpretation:95% of all the confidence intervals that can be constructed will contain the unknown true parameterA specific interval eithe

16、r will contain or will not contain the true parameterNo 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 Pr

17、entice-Hall, Inc.Chap 8-19Confidence Interval for ( Known) AssumptionsPopulation standard deviation is knownPopulation is normally distributedIf population is not normal, use large sampleConfidence interval estimate: where is the point estimate Z/2 is the normal distribution critical value for a pro

18、bability of /2 in each tail is the standard error Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-20Finding the Critical Value, Z/2Consider a 95% confidence interval:Z/2 = -1.96Z/2 = 1.96Point EstimateLower Confidence LimitUpperConfidence LimitZ units:X units:Point Estimate0Basic Busin

19、ess Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-21Common Levels of ConfidenceCommonly 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 200

20、9 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 8-23ExampleA sample of 11 circuits

21、 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. Determine a 95% confidence interval for the true mean resistance of the population.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-24Exampl

22、eA 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. Solution:(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-25InterpretationWe are 95% confident that the t

23、rue mean resistance is between 1.9932 and 2.4068 ohms Although 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 UnknownConfidenceInter

24、valsPopulationProportion KnownBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-27Do You Ever Truly Know ?Probably not!In virtually all real world business situations, is not known.If there is a situation where is known then is also known (since to calculate you need to know .)If you tru

25、ly know there would be no need to gather a sample to estimate it.Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-28If the population standard deviation is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to s

26、ampleSo we use the t distribution instead of the normal distributionConfidence Interval for ( Unknown) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-29AssumptionsPopulation standard deviation is unknownPopulation is normally distributedIf population is not normal, use large sampleUse

27、 Students t DistributionConfidence Interval Estimate:(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 Distributi

28、onThe t is a family of distributionsThe t/2 value depends on degrees of freedom (d.f.)Number 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 b

29、e 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 mea

30、n 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 increase

31、sBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-33Students t TableUpper Tail Areadf.25.10.0511.0003.0786.31420.8171.8862.92030.7651.6382.353t02.920The 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, 11

32、e 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

33、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 d.f. = n 1 = 24, soThe confidence interval is 46.698 53.302Basic Business Statistics, 11e 2009 Prentice-Hall, Inc

34、.Chap 8-36Example of t distribution confidence intervalInterpreting this interval requires the assumption that the population you are sampling from is approximately a normal distribution (especially since n is only 25).This condition can be checked by creating a:Normal probability plot orBoxplot(con

35、tinued)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, An interval estimate for

36、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, Recall that the distribution of the sample proportion is approxima

37、tely normal if the sample size is large, with standard deviationWe will estimate this with sample data:(continued)Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-40Confidence Interval EndpointsUpper and lower confidence limits for the population proportion are calculated with the formu

38、lawhere Z/2 is the standard normal value for the level of confidence desiredp is the sample proportionn is the sample sizeNote: must have np 5 and n(1-p) 5Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-41ExampleA random sample of 100 people shows that 25 are left-handed. Form a 95% co

39、nfidence interval for the true proportion of left-handersBasic Business Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-42ExampleA 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, 11

40、e 2009 Prentice-Hall, Inc.Chap 8-43InterpretationWe are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although 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 t

41、his 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 ErrorThe required sample size can be found to

42、 reach a desired margin of error (e) with a specified level of confidence (1 - )The margin of error is also called sampling errorthe amount of imprecision in the estimate of the population parameterthe amount added and subtracted to the point estimate to form the confidence intervalBasic Business St

43、atistics, 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 Busi

44、ness Statistics, 11e 2009 Prentice-Hall, Inc.Chap 8-48Determining Sample SizeTo determine the required sample size for the mean, you must know:The desired level of confidence (1 - ), which determines the critical value, Z/2The acceptable sampling error, eThe standard deviation, (continued)Basic Busi

45、ness 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 unknow

46、nIf unknown, can be estimated when using the required sample size formulaUse a value for that is expected to be at least as large as the true Select 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 SizeTo determine the required sample size for the proportion, you must know:The desired leve