[教育]應(yīng)用統(tǒng)計(jì)學(xué)英文課件businessstatisticsch08confidenceintervalestima

1、Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.,Chap 8-1,Chapter 8Confidence Interval Estimation,Business Statistics:A First Course5th Edition,Chapter ProblemSaxon Home Improvement,Saxon Home Improve

2、ment,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-4,Learning Objectives,In this chapter, you learn: To construct and interpret confidence interval estimates for the mean and the proportionHow

3、to determine the sample size necessary to develop a confidence interval for the mean or proportion,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-5,Chapter Outline,Content of this chapterConfiden

4、ce Intervals for the Population Mean, μwhen Population Standard Deviation σ is Knownwhen Population Standard Deviation σ is UnknownConfidence Intervals for the Population Proportion, πDetermining the Required Sample

5、Size,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-6,Point and Interval Estimates,A point estimate is a single number a confidence interval provides additional information about the variability

6、of the estimate,,,,,,Point Estimate,Lower Confidence Limit,UpperConfidence Limit,,Width of confidence interval,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-7,We can estimate a Population P

7、arameter …,,,,Point Estimates,with a SampleStatistic(a Point Estimate),Mean,Proportion,,,,,,,,,p,π,X,,μ,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-8,Confidence Intervals,How much uncertainty

8、 is associated with a point estimate of a population parameter?An interval estimate provides more information about a population characteristic than does a point estimateSuch interval estimates are called confidence

9、intervals,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-9,Confidence Interval Estimate,An interval gives a range of values:Takes into consideration variation in sample statistics from sample to

10、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,Basic Bus

11、iness Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-10,Confidence Interval Example,Cereal fill example Population has µ = 368 and σ = 15. If you take a sample of size n = 25 you know368 ± 1.96

12、* 15 / = (362.12, 373.88) contains 95% of the sample meansWhen you don’t know µ, you use X to estimate µIf X = 362.3 the interval is 362.3 ± 1.96 * 15 / = (356.42, 368.18)Since 356.42 ≤

13、81; ≤ 368.18, the interval based on this sample makes a correct statement about µ.But what about the intervals from other possible samples of size 25?,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, In

14、c..,Chap 8-11,Confidence Interval Example,(continued),,Point and Interval Estimates,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-13,Confidence Interval Example,In practice you only take one samp

15、le of size nIn practice you do not know µ so you do not know if the interval actually contains µHowever you do know that 95% of the intervals formed in this manner will contain µThus, based on the one s

16、ample, you actually selected you can be 95% confident your interval will contain µ (this is a 95% confidence interval),(continued),Note: 95% confidence is based on the fact that we used Z = 1.96.,Basic Business Sta

17、tistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-14,Estimation Process,,(mean, μ, is unknown),Population,Random Sample,,,Mean X = 50,,,Sample,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Ch

18、ap 8-15,General Formula,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

19、 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 estimate,Basic Business Statisti

20、cs, 11e © 2009 Prentice-Hall, Inc..,Chap 8-16,Confidence Level,Confidence LevelThe confidence that the interval will contain the unknown population parameterA percentage (less than 100%),Basic Business Statistics,

21、 11e © 2009 Prentice-Hall, Inc..,Chap 8-17,Confidence Level, (1-?),Suppose confidence level = 95% Also written (1 - ?) = 0.95, (so ? = 0.05)A relative frequency interpretation:95% of all the confidence interval

22、s 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 specific interval,(continued),Basic Business S

23、tatistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-18,Confidence Intervals,,Population Mean,,σ Unknown,,Confidence,Intervals,,PopulationProportion,,σ Known,,,,,,,,,,Basic Business Statistics, 11e © 2009 Prenti

24、ce-Hall, Inc..,Chap 8-19,Confidence Interval for μ(σ Known),AssumptionsPopulation standard deviation σ is knownPopulation is normally distributedIf population is not normal, use large sampleConfidence interval esti

25、mate: where is the point estimate Zα/2 is the normal distribution critical value for a probability of ?/2 in each tail is the standard error,Basic Business Statistics, 11e © 2009 P

26、rentice-Hall, Inc..,Chap 8-20,,Finding the Critical Value, Zα/2,Consider a 95% confidence interval:,,,,,,,,,,,,,,,,,Zα/2 = -1.96,Zα/2 = 1.96,,,Point Estimate,Lower Confidence Limit,UpperConfidence Limit,Z units:,X un

27、its:,Point Estimate,0,,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-21,,Common Levels of Confidence,Commonly used confidence levels are 90%, 95%, and 99%,Confidence Level,Confidence Coefficien

28、t,,Zα/2 value,1.281.6451.962.332.583.083.27,0.800.900.950.980.990.9980.999,80%90%95%98%99%99.8%99.9%,,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-22,,,,,,,,Intervals and Lev

29、el of Confidence,Confidence Intervals,,Intervals extend from to,(1-?)x100%of intervals constructed contain μ; (?)x100% do not.,,,,,,Sampling Distribution of the Mean,,,,,,,,,,,x,,,x1,,,x2,,,,,,,,Basic Business St

30、atistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-23,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 deviatio

31、n 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 8-24,,Example,A sample of 11 circuits from a lar

32、ge 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, I

33、nc..,Chap 8-25,Interpretation,We are 95% 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, 95% of intervals formed in this manner will

34、 contain the true mean,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-26,,Confidence Intervals,,Population Mean,,σ Unknown,,Confidence,Intervals,,PopulationProportion,σ Known,,,,,,,,,,Basic Busi

35、ness Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-27,Do You Ever Truly Know σ?,Probably not!In virtually all real world business situations, σ is not known.If there is a situation where σ is known then

36、81; is also known (since to calculate σ you need to know µ.)If you truly know µ there would be no need to gather a sample to estimate it.,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8

37、-28,If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sampleSo we use the t distribution ins

38、tead of the normal distribution,Confidence Interval for μ(σ Unknown),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-29,AssumptionsPopulation standard deviation is unknownPopulation is normally

39、distributedIf population is not normal, use large sampleUse Student’s t DistributionConfidence Interval Estimate:(where tα/2 is the critical value of the t distribution with n -1 degrees of freedom and an area of

40、α/2 in each tail),Confidence Interval for μ(σ Unknown),(continued),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-30,,Student’s t Distribution,,The t is a family of distributionsThe tα/2 value d

41、epends on degrees of freedom (d.f.)Number of observations that are free to vary after sample mean has been calculatedd.f. = n - 1,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-31,If the mea

42、n 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

43、 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 Bus

44、iness Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-32,,,Student’s t Distribution,,,,,t,,,,,,,,,,,,,,0,,t (df = 5),,t (df = 13),t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the

45、normal,,Standard Normal(t with df = ∞),,,Note: t Z as n increases,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-33,,Student’s t Table,,Upper Tail Area,,df,.25,,.10,,.05,,1,1.000,3.078

46、,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 ? = 0.10 ?/2 = 0.05,?/2 = 0.05,,,,,,,,,

47、Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-34,,Selected t distribution values,With comparison to the Z value,Confidence t t t Z Level

48、 (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

49、 2.042 1.96 0.99 3.169 2.845 2.750 2.58,,,,Note: t Z as n increases,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-35,,,Example of t dist

50、ribution confidence interval,A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μd.f. = n – 1 = 24, soThe confidence interval is,,46.698 ≤ μ ≤ 53.302,Basic Business Statistics, 11e

51、 © 2009 Prentice-Hall, Inc..,Chap 8-36,Example of t distribution confidence interval,Interpreting this interval requires the assumption that the population you are sampling from is approximately a normal distributio

52、n (especially since n is only 25).This condition can be checked by creating a:Normal probability plot orBoxplot,(continued),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-37,,,Confidence Interv

53、als,,Population Mean,,σ Unknown,,Confidence,Intervals,PopulationProportion,σ Known,,,,,,,,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-38,Confidence Intervals for the Population Proportion,

54、 π,An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-3

55、9,Confidence Intervals for the Population Proportion, π,Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviationWe will estimate this with sa

56、mple data:,(continued),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-40,Confidence Interval Endpoints,Upper and lower confidence limits for the population proportion are calculated with the formu

57、lawhere Zα/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,Basic Business Statistics, 11e 

58、9; 2009 Prentice-Hall, Inc..,Chap 8-41,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,Basic Business Statistics, 11e © 20

59、09 Prentice-Hall, Inc..,Chap 8-42,,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),Basic Business Statistics, 11e &

60、#169; 2009 Prentice-Hall, Inc..,Chap 8-43,Interpretation,We 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

61、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 8-44,,Determi

62、ning Sample Size,,For the Mean,,Determining,Sample Size,For theProportion,,,,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-45,Sampling Error,The required sample size can be found to reach a de

63、sired 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 subtrac

64、ted to the point estimate to form the confidence interval,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-46,Determining Sample Size,,For the Mean,,Determining,Sample Size,,,,,,Sampling error (mar

65、gin of error),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-47,Determining Sample Size,,For the Mean,,Determining,Sample Size,,,,,(continued),Now solve for n to get,,,Basic Business Statistics

66、, 11e © 2009 Prentice-Hall, Inc..,Chap 8-48,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

67、α/2The acceptable sampling error, eThe standard deviation, σ,(continued),Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-49,Required Sample Size Example,If ? = 45, what sample size is needed to e

68、stimate the mean within ± 5 with 90% confidence?,(Always round up),So the required sample size is n = 220,,Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..,Chap 8-50,If σ is unknown,If unknown, σ ca

69、n be estimated when using the required sample size formulaUse a value for σ that is expected to be at least as large as the true σSelect a pilot sample and estimate σ with the sample standard deviation, S,Basic B