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1、Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-1,Business Statistics:A Decision-Making Approach6th Edition,Chapter 7Estimating Population Values,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-2,Chapter Goals,After completing this ch
2、apter,you should be able to:Distinguish between a point estimate and a confidence interval estimateConstruct and interpret a confidence interval estimate for a single population mean using both the z and t distributionsDetermine the required sample size to estimate a single population mean within a
3、specified margin of errorForm and interpret a confidence interval estimate for a single population proportion,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-3,Confidence Intervals,Content of this chapterConfidence Intervals for the Population Mean,when Population St
4、andard Deviation is Knownwhen Population Standard Deviation is UnknownDetermining the Required Sample SizeConfidence Intervals for the Population Proportion,p,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-4,Point and Interval Estimates,A point estimate is a single
5、number,a confidence interval provides additional information about variability,Point Estimate,Lower Confidence Limit,UpperConfidence Limit,Width of confidence interval,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-5,Point Estimates,We can estimate a Population Para
6、meter,with a SampleStatistic(a Point Estimate),Mean,Proportion,p,p,x,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-6,Confidence Intervals,How much uncertainty is associated with a point estimate of a population parameter?An interval estimate provides more informati
7、on about a population characteristic than does a point estimateSuch interval estimates are called confidence intervals,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-7,Confidence Interval Estimate,An interval gives a range of values:Takes into consideration variatio
8、n in sample statistics from sample to sampleBased on observation from 1 sampleGives information about closeness to unknown population parametersStated in terms of level of confidenceNever 100%sure,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-8,Estimation Process,(
9、mean,is unknown),Population,Random Sample,Mean x=50,Sample,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-9,General Formula,The general formula for all confidence intervals is:,Point Estimate(Critical Value)(Standard Error),Business Statistics:A Decision-Making Appr
10、oach,6e 2005 Prentice-Hall,Inc.,Chap 7-10,Confidence Level,Confidence LevelConfidence in which the interval will contain the unknown population parameterA percentage(less than 100%),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-11,Confidence Level,(1-),Suppose conf
11、idence level=95%Also written(1-)=.95A relative frequency interpretation:In the long run,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 involved in a specifi
12、c interval,(continued),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-12,Confidence Intervals,Population Mean,Unknown,Confidence,Intervals,PopulationProportion,Known,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-13,Confidence Inter
13、val for(Known),AssumptionsPopulation standard deviation is knownPopulation is normally distributedIf population is not normal,use large sampleConfidence interval estimate,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-14,Finding the Critical Value,Consider a 95%conf
14、idence interval:,z.025=-1.96,z.025=1.96,Point Estimate,Lower Confidence Limit,UpperConfidence Limit,z units:,x units:,Point Estimate,0,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-15,Common Levels of Confidence,Commonly used confidence levels are 90%,95%,and 99%,C
15、onfidence Level,Confidence Coefficient,z value,1.281.6451.962.332.573.083.27,.80.90.95.98.99.998.999,80%90%95%98%99%99.8%99.9%,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-16,Interval and Level of Confidence,Confidence Intervals,Intervals extend from to,100(1-)%of
16、 intervals constructed contain;100%do not.,Sampling Distribution of the Mean,x,x1,x2,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-17,Margin of Error,Margin of Error(e):the amount added and subtracted to the point estimate to form the confidence interval,Example:Ma
17、rgin of error for estimating,known:,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-18,Factors Affecting Margin of Error,Data variation,:e as Sample size,n:e as n Level of confidence,1-:e if 1-,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.
18、,Chap 7-19,Example,A 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.35 ohms.Determine a 95%confidence interval for the true mean resistance of the population.,Business Statistics:A Decision-M
19、aking Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-20,Example,A 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.35 ohms.Solution:,(continued),Business Statistics:A Decision-Making Approach,6e 20
20、05 Prentice-Hall,Inc.,Chap 7-21,Interpretation,We are 98%confident that the true mean resistance is between 1.9932 and 2.4068 ohms Although the true mean may or may not be in this interval,98%of intervals formed in this manner will contain the true meanAn incorrect interpretation is that there is 98
21、%probability that this interval contains the true population mean.(This interval either does or does not contain the true mean,there is no probability for a single interval),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-22,Confidence Intervals,Population Mean,Unkno
22、wn,Confidence,Intervals,PopulationProportion,Known,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-23,If the population standard deviation is unknown,we can substitute the sample standard deviation,s This introduces extra uncertainty,since s is variable from sample t
23、o sampleSo we use the t distribution instead of the normal distribution,Confidence Interval for(Unknown),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-24,AssumptionsPopulation standard deviation is unknownPopulation is normally distributedIf population is not norma
24、l,use large sampleUse Students t DistributionConfidence Interval Estimate,Confidence Interval for(Unknown),(continued),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-25,Students t Distribution,The t is a family of distributionsThe t value depends on degrees of freed
25、om(d.f.)Number of observations that are free to vary after sample mean has been calculatedd.f.=n-1,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-26,If 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),Idea:N
26、umber of observations that are free to vary after sample mean has been calculatedExample:Suppose the mean of 3 numbers is 8.0Let x1=7Let x2=8What is x3?,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),Business Statistics:A Deci
27、sion-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-27,Students t Distribution,t,0,t(df=5),t(df=13),t-distributions are bell-shaped and symmetric,but have fatter tails than the normal,Standard Normal(t with df=),Note:t z as n increases,Business Statistics:A Decision-Making Approach,6e 2005 Prenti
28、ce-Hall,Inc.,Chap 7-28,Students t Table,Upper Tail Area,df,.25,.10,.05,1,1.000,3.078,6.314,2,0.817,1.886,2.920,3,0.765,1.638,2.353,t,0,2.920,The body of the table contains t values,not probabilities,Let:n=3 df=n-1=2=.10/2=.05,/2=.05,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hal
29、l,Inc.,Chap 7-29,t distribution values,With comparison to the z value,Confidence t t t z Level(10 d.f.)(20 d.f.)(30 d.f.)_.80 1.372 1.325 1.310 1.28.90 1.812 1.725 1.697 1.64.95 2.228 2.086 2.042 1.96.99 3.169 2.845 2.750 2.57,Note:t z as n increases,Business Statistics:A Decision-Making Approach,6e
30、 2005 Prentice-Hall,Inc.,Chap 7-30,Example,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,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-31,Approximation for Large Samples,Since t app
31、roaches z as the sample size increases,an approximation is sometimes used when n 30:,Technically correct,Approximation for large n,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-32,Determining Sample Size,The required sample size can be found to reach a desired marg
32、in of error(e)and level of confidence(1-)Required sample size,known:,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-33,Required Sample Size Example,If=45,what sample size is needed to be 90%confident of being correct within 5?,(Always round up),So the required sampl
33、e size is n=220,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-34,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 sa
34、mple standard deviation,s,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-35,Confidence Intervals,Population Mean,Unknown,Confidence,Intervals,PopulationProportion,Known,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-36,Confidence In
35、tervals for the Population Proportion,p,An interval estimate for the population proportion(p)can be calculated by adding an allowance for uncertainty to the sample proportion(p),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-37,Confidence Intervals for the Populatio
36、n Proportion,p,Recall that the distribution of the sample proportion is approximately normal if the sample size is large,with standard deviationWe will estimate this with sample data:,(continued),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-38,Confidence interval
37、endpoints,Upper and lower confidence limits for the population proportion are calculated with the formulawhere z is the standard normal value for the level of confidence desiredp is the sample proportionn is the sample size,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Ch
38、ap 7-39,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,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-40,Example,A random sample of 100 people shows that 25 are left-handed.Fo
39、rm a 95%confidence interval for the true proportion of left-handers.,1.,2.,3.,(continued),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-41,Interpretation,We are 95%confident that the true percentage of left-handers in the population is between 16.51%and 33.49%.Alth
40、ough this range 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.,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-42,Changing the sample size,Increases in the sample size reduce
41、 the width of the confidence interval.Example:If the sample size in the above example is doubled to 200,and if 50 are left-handed in the sample,then the interval is still centered at.25,but the width shrinks to.19.31,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-43
42、,Finding the Required Sample Size for proportion problems,Solve for n:,Define the margin of error:,p can be estimated with a pilot sample,if necessary(or conservatively use p=.50),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-44,What sample size.?,How large a sampl
43、e would be necessary to estimate the true proportion defective in a large population within 3%,with 95%confidence?(Assume a pilot sample yields p=.12),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-45,What sample size.?,Solution:For 95%confidence,use Z=1.96E=.03p=.1
44、2,so use this to estimate p,So use n=451,(continued),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-46,Using PHStat,PHStat can be used for confidence intervals for the mean or proportiontwo options for the mean:known and unknown population standard deviationrequired
45、 sample size can also be found,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-47,PHStat Interval Options,options,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-48,PHStat Sample Size Options,Business Statistics:A Decision-Making Appr
46、oach,6e 2005 Prentice-Hall,Inc.,Chap 7-49,Using PHStat(for,unknown),A random sample of n=25 has x=50 and s=8.Form a 95%confidence interval for,Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-50,Using PHStat(sample size for proportion),How large a sample would be nece
47、ssary to estimate the true proportion defective in a large population within 3%,with 95%confidence?(Assume a pilot sample yields p=.12),Business Statistics:A Decision-Making Approach,6e 2005 Prentice-Hall,Inc.,Chap 7-51,Chapter Summary,Illustrated estimation processDiscussed point estimatesIntroduced interval estimatesDiscussed confidence interval estimation for the mean(known)Addressed determining sample sizeDiscussed confidence interval estimation for the mean(unknown)Discussed confidence interval estimation for the proportion,
