商务统计学英文版教学课件第8章.ppt

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1、Confidence Interval Estimation,Chapter 8,Confidence Interval Estimation,Objectives,In this chapter, you learn: To construct and interpret confidence interval estimates for the population mean and the population proportionTo determine the sample size necessary to develop a confidence interval for the

2、 population mean or population proportion,ObjectivesIn this chapter, you,Chapter Outline,Content of this chapterConfidence Intervals for the Population Mean, when Population Standard Deviation is Knownwhen Population Standard Deviation is UnknownConfidence Intervals for the Population Proportion, De

3、termining the Required Sample Size,Chapter OutlineContent of this,Point and Interval Estimates,A point estimate is a single number, a confidence interval provides additional information about the variability of the estimate,Point Estimate,Lower Confidence Limit,UpperConfidence Limit,Width of confide

4、nce interval,DCOVA,Point and Interval EstimatesA,We can estimate a Population Parameter ,Point Estimates,with a SampleStatistic(a Point Estimate),Mean,Proportion,p,X,DCOVA,We can estimate a Point Estima,Confidence Intervals,How much uncertainty is associated with a point estimate of a population par

5、ameter?An interval estimate provides more information about a population characteristic than does a point estimateSuch interval estimates are called confidence intervals,DCOVA,Confidence IntervalsHow much u,Confidence Interval Estimate,An interval gives a range of values:Takes into consideration var

6、iation in sample 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% confident,DCOVA,Confidence Interval EstimateAn,Confidence Int

7、erval Example,Cereal 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). 95% of the intervals formed in this manner will contain .When you dont know , you use X to estimate If X = 362.3 the interval is 362.3 1.96 * 15 / = (356.4

8、2, 368.18)Since 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?,DCOVA,Confidence Interval ExampleCer,Confidence Interval Example,(continued),DCOVA,Confidence Interval Example(co,Confidence Interval

9、Example,In practice you only take one sample of size nIn practice you do not know so you 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 i

10、nterval will contain (this is a 95% confidence interval),(continued),Note: 95% confidence is based on the fact that we used Z = 1.96.,DCOVA,Confidence Interval ExampleIn,Estimation Process,(mean, , is unknown),Population,Random Sample,Mean X = 50,Sample,DCOVA,Estimation Process(mean, , is,General Fo

11、rmula,The 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 desi

12、red confidence levelStandard Error is the standard deviation of the point estimate,DCOVA,General FormulaThe general for,Confidence Level,Confidence the interval will contain the unknown population parameterA percentage (less than 100%),DCOVA,Confidence LevelConfidence the,Confidence Level, (1-),Supp

13、ose 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 either will contain or will not contain the true parameterNo probability involv

14、ed in a specific interval,(continued),DCOVA,Confidence Level, (1-)Suppose,Confidence Intervals,DCOVA,Confidence IntervalsPopulation,Confidence Interval for ( Known),AssumptionsPopulation standard deviation is knownPopulation is normally distributedIf population is not normal, use large sample (n 30)

15、Confidence interval estimate: 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,DCOVA,Confidence Interval for (,Finding the Critical Value, Z/2,Consider a 95% confidence interval:,Z/2 = -1.96,Z/2 = 1.96,Point Estimate

16、,Lower Confidence Limit,UpperConfidence Limit,Z units:,X units:,Point Estimate,0,DCOVA,Finding the Critical Value, Z,Common Levels of Confidence,Commonly used confidence levels are 90%, 95%, and 99%,Confidence Level,Confidence Coefficient,Z/2 value,1.281.6451.962.332.583.083.27,0.800.900.950.980.990

17、.9980.999,80%90%95%98%99%99.8%99.9%,DCOVA,Common Levels of ConfidenceCom,Intervals and Level of Confidence,Confidence Intervals,Intervals extend from to,(1-)100%of intervals constructed contain ; ()100% do not.,Sampling Distribution of the Mean,x,x1,x2,DCOVA,Intervals and Level of Confide,Example,A

18、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. Determine a 95% confidence interval for the true mean resistance of the population.,DCOVA,ExampleA sample of 11 circuits,Example,A s

19、ample 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),DCOVA,ExampleA sample of 11 circuits,Interpretation,We are 95% confident that the true mean resistance is betwe

20、en 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 mean,DCOVA,InterpretationWe are 95% confi,Confidence Intervals,DCOVA,Confidence IntervalsPopulation,Do You Ever Truly Know ?,Probably not!In virtually all

21、 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 truly know there would be no need to gather a sample to estimate it.,Do You Ever Truly Know ?Proba,If the population standard deviation is unknown, we

22、 can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sampleSo we use the t distribution instead of the normal distribution,Confidence Interval for ( Unknown),DCOVA,If the population standard dev,AssumptionsPopulation standard deviatio

23、n is unknownPopulation is normally distributedIf population is not normal, use large sample (n 30)Use 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 ( Unk

24、nown),(continued),DCOVA,AssumptionsConfidence Interval,Students t Distribution,The 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 - 1,DCOVA,Students t DistributionThe t,If the

25、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 sam

26、ple mean has been calculatedExample: Suppose the mean of 3 numbers is 8.0 Let X1 = 7Let X2 = 8What is X3?,DCOVA,If the mean of these three val,Students t Distribution,Note: t Z as n increases,DCOVA,Students t Distributiont0t (,Students t Table,DCOVA,Students t TableDCOVAUpper Ta,Selected t distribut

27、ion values,With comparison to the Z value,Confidence 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 2.58,Note: t Z as n increases,DCOVA,Selected t distribution values,Example of t distrib

28、ution 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 53.302,DCOVA,Example of t distribution conf,Example of t distribution confidence interval,Interpreting this interval requires the assumpti

29、on 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,(continued),DCOVA,Example of t distribution conf,Confidence Intervals,DCOVA,Confidence IntervalsPopulation

30、,Confidence Intervals for the Population Proportion, ,An interval estimate for the population proportion ( ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ),DCOVA,Confidence Intervals for the,Confidence Intervals for the Population Proportion, ,Recall that the

31、 distribution of the sample proportion is approximately normal if the sample size is large, with standard deviationWe will estimate this with sample data:,(continued),DCOVA,Confidence Intervals for the,Confidence Interval Endpoints,Upper and lower confidence limits for the population proportion are

32、calculated with the formulawhere 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) 5,DCOVA,Confidence Interval EndpointsU,Example,A random sample of 100 people shows that 25 are left-handed. Form a 95% co

33、nfidence interval for the true proportion of left-handers,DCOVA,ExampleA random sample of 100,Example,A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.,(continued),DCOVA,ExampleA random sample of 100,Interpretation,We

34、 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 this manner will contain the true proportion.,DCOVA,

35、InterpretationWe are 95% confi,Determining Sample Size,DCOVA,Determining Sample SizeFor the,Sampling Error,The required sample size can be found to 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 imprecisio

36、n in the estimate of the population parameterthe amount added and subtracted to the point estimate to form the confidence interval,DCOVA,Sampling ErrorThe required sam,Determining Sample Size,For the Mean,Determining,Sample Size,Sampling error (margin of error),DCOVA,Determining Sample SizeFor the,D

37、etermining Sample Size,For the Mean,Determining,Sample Size,(continued),Now solve for n to get,DCOVA,Determining Sample SizeFor the,Determining Sample Size,To determine the required sample size for the mean, you must know:The desired level of confidence (1 - ), which determines the critical value, Z

38、/2The acceptable sampling error, eThe standard deviation, ,(continued),DCOVA,Determining Sample SizeTo dete,Required Sample Size Example,If = 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 = 220,DCOVA,Required Samp

39、le Size ExampleIf,If is unknown,If 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,DCOVA,If is unknownIf unknown, ,Determining Sample Siz

40、e,Determining,Sample Size,For theProportion,Now solve for n to get,(continued),DCOVA,Determining Sample SizeDetermi,Determining Sample Size,To determine the required sample size for the proportion, you must know:The desired level of confidence (1 - ), which determines the critical value, Z/2The acce

41、ptable sampling error, eThe true proportion of events of interest, can be estimated with a pilot sample if necessary (or conservatively use 0.5 as an estimate of ),(continued),DCOVA,Determining Sample SizeTo dete,Required Sample Size Example,How large a sample would be necessary to estimate the true

42、 proportion of defectives in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p = 0.12),DCOVA,Required Sample Size ExampleHo,Required Sample Size Example,Solution:For 95% confidence, use Z/2 = 1.96e = 0.03p = 0.12, so use this to estimate ,So use n = 451,(continued),D

43、COVA,Required Sample Size ExampleSo,Ethical Issues,A confidence interval estimate (reflecting sampling error) should always be included when reporting a point estimate The level of confidence should always be reported The sample size should be reportedAn interpretation of the confidence interval est

44、imate should also be provided,Ethical IssuesA confidence int,Chapter Summary,In this chapter we discussed: The construction and interpretation of confidence interval estimates for the population mean and the population proportionThe determination of the sample size necessary to develop a confidence

45、interval for the population mean or population proportion,Chapter SummaryIn this chapter,On Line Topic: Bootstrapping,Chapter 8,On Line Topic: BootstrappingC,Bootstrapping Is A Method To Use When Population Is Not Normal,To estimate a population mean using bootstrapping, you would:Select a random sa

46、mple of size n without replacement from a population of size N.Resample the initial sample by selecting n values with replacement from the initial sample.Compute X from this resample.Repeat steps 2 & 3 m different times.Construct the resampling distribution of X.Construct an ordered array of the ent

47、ire set of resampled Xs.In this ordered array find the value that cuts off the smallest /2(100%) and the value that cuts off the largest /2(100%). These values provide the lower and upper limits of the bootstrap confidence interval estimate of .,DCOVA,Bootstrapping Is A Method To U,Bootstrapping Req

48、uires The Use of Software As Minitab or JMP,Typically a very large number (thousands) of resamples are used.Software is needed to:Automate the resampling processCalculate the appropriate sample statisticCreate the ordered arrayFind the lower and upper confidence limits,DCOVA,Bootstrapping Requires T

49、he Use,Bootstrapping Example - Processing Time of Life Insurance Applications,DCOVA,73 19 16 64 28 28 31 90 60 56 31 56 22 18 45 48 17 17 17 91 92 63 50 51 69 16 17,Sample of 27 times taken without replacement from population,From boxplot conclude population is not normal so t confidence interval is

50、 not appropriate.Use bootstrapping to form a confidence interval for .,Bootstrapping Example - Pro,Comparing the original sample to the first resample with replacement,DCOVA,73 19 16 64 28 28 31 90 60 56 31 56 22 18 45 48 17 17 17 91 92 63 50 51 69 16 17,Sample of 27 times taken without replacement