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### Chap07

1. 1. Chap 7-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.Chapter 7Sampling andSampling DistributionsStatistics forBusiness and Economics6thEdition
2. 2. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-2Chapter GoalsAfter completing this chapter, you should be able to: Describe a simple random sample and why sampling isimportant Explain the difference between descriptive andinferential statistics Define the concept of a sampling distribution Determine the mean and standard deviation for thesampling distribution of the sample mean, Describe the Central Limit Theorem and its importance Determine the mean and standard deviation for thesampling distribution of the sample proportion, Describe sampling distributions of sample variancespˆX
3. 3. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-3 Descriptive statistics Collecting, presenting, and describing data Inferential statistics Drawing conclusions and/or making decisionsconcerning a population based only onsample dataTools of Business Statistics
4. 4. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-4 A Population is the set of all items or individualsof interest Examples: All likely voters in the next electionAll parts produced todayAll sales receipts for November A Sample is a subset of the population Examples: 1000 voters selected at random for interviewA few parts selected for destructive testingRandom receipts selected for auditPopulations and Samples
5. 5. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-5Population vs. Samplea b c def gh i jk l m no p q rs t u v wx y zPopulation Sampleb cg i no r uy
6. 6. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-6Why Sample? Less time consuming than a census Less costly to administer than a census It is possible to obtain statistical results of asufficiently high precision based on samples.
7. 7. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-7Simple Random Samples Every object in the population has an equal chance ofbeing selected Objects are selected independently Samples can be obtained from a table of randomnumbers or computer random number generators A simple random sample is the ideal against whichother sample methods are compared
8. 8. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-8 Making statements about a population byexamining sample resultsSample statistics Population parameters(known) Inference (unknown, but canbe estimated fromsample evidence)Sample PopulationInferential Statistics
9. 9. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-9Inferential Statistics Estimation e.g., Estimate the population meanweight using the sample meanweight Hypothesis Testing e.g., Use sample evidence to testthe claim that the population meanweight is 120 poundsDrawing conclusions and/or making decisionsconcerning a population based on sample results.
10. 10. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-10Sampling Distributions A sampling distribution is a distributionof all of the possible values of a statisticfor a given size sample selected from apopulation
11. 11. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-11Chapter OutlineSamplingDistributionsSamplingDistribution ofSampleMeanSamplingDistribution ofSampleProportionSamplingDistribution ofSampleVariance
12. 12. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-12Sampling Distributions ofSample MeansSamplingDistributionsSamplingDistribution ofSampleMeanSamplingDistribution ofSampleProportionSamplingDistribution ofSampleVariance
13. 13. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-13Developing aSampling Distribution Assume there is a population … Population size N=4 Random variable, X,is age of individuals Values of X:18, 20, 22, 24 (years)A B C D
14. 14. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-14.25018 20 22 24A B C DUniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing aSampling Distribution21424222018NXμ i=+++==∑2.236Nμ)(Xσ2i=−=∑
15. 15. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-151st2ndObservationObs 18 20 22 2418 18,18 18,20 18,22 18,2420 20,18 20,20 20,22 20,2422 22,18 22,20 22,22 22,2424 24,18 24,20 24,22 24,2416 possible samples(sampling withreplacement)Now consider all possible samples of size n = 21st 2nd ObservationObs 18 20 22 2418 18 19 20 2120 19 20 21 2222 20 21 22 2324 21 22 23 24(continued)Developing aSampling Distribution16 SampleMeans
16. 16. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-161st 2nd ObservationObs 18 20 22 2418 18 19 20 2120 19 20 21 2222 20 21 22 2324 21 22 23 24Sampling Distribution of All Sample Means18 19 20 21 22 23 240.1.2.3P(X)XSample MeansDistribution16 Sample Means_Developing aSampling Distribution(continued)(no longer uniform)_
17. 17. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-17Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)μ211624211918NX)XE( i==++++==∑ 1.581621)-(2421)-(1921)-(18Nμ)X(σ2222iX=+++=−=∑
18. 18. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-18Comparing the Population with itsSampling Distribution18 19 20 21 22 23 240.1.2.3P(X)X18 20 22 24A B C D0.1.2.3PopulationN = 4P(X)X _1.58σ21μ XX==2.236σ21μ ==Sample Means Distributionn = 2_
19. 19. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-19Expected Value of Sample Mean Let X1, X2, . . . Xn represent a random sample from apopulation The sample mean value of these observations isdefined as∑==n1iiXn1X
20. 20. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-20Standard Error of the Mean Different samples of the same size from the samepopulation will yield different sample means A measure of the variability in the mean from sample tosample is given by the Standard Error of the Mean: Note that the standard error of the mean decreases asthe sample size increasesnσσX=
21. 21. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-21If the Population is Normal If a population is normal with mean μ andstandard deviation σ, the sampling distributionof is also normally distributed withandXμμX=nσσX=
22. 22. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-22Z-value for Sampling Distributionof the Mean Z-value for the sampling distribution of :where: = sample mean= population mean= population standard deviationn = sample sizeXμσnσμ)X(σμ)X(ZX−=−=X
23. 23. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-23Finite Population Correction Apply the Finite Population Correction if: a population member cannot be included morethan once in a sample (sampling is withoutreplacement), and the sample is large relative to the population(n is greater than about 5% of N) Thenor1NnNnσσX−−=1NnNnσ)XVar(2−−=
24. 24. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-24Finite Population Correction If the sample size n is not small compared to thepopulation size N , then use1NnNnσμ)X(Z−−−=
25. 25. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-25Normal PopulationDistributionNormal SamplingDistribution(has the same mean)Sampling Distribution Properties(i.e. is unbiased )xxxμμx =μxμ
26. 26. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-26Sampling Distribution Properties For sampling with replacement:As n increases,decreasesLargersample sizeSmallersample sizex(continued)xσμ
27. 27. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-27If the Population is not Normal We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will beapproximately normal as long as the sample size islarge enough.Properties of the sampling distribution:andμμx =nσσx =
28. 28. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-28n↑Central Limit TheoremAs thesamplesize getslargeenough…the samplingdistributionbecomesalmost normalregardless ofshape ofpopulationx
29. 29. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-29Population DistributionSampling Distribution(becomes normal as n increases)Central TendencyVariationxxLargersamplesizeSmallersample sizeIf the Population is not Normal(continued)Sampling distributionproperties:μμx =nσσx =xμμ
30. 30. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-30How Large is Large Enough? For most distributions, n > 25 will give asampling distribution that is nearly normal For normal population distributions, thesampling distribution of the mean is alwaysnormally distributed
31. 31. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-31Example Suppose a population has mean μ = 8 andstandard deviation σ = 3. Suppose a randomsample of size n = 36 is selected. What is the probability that the sample mean isbetween 7.8 and 8.2?
32. 32. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-32ExampleSolution: Even if the population is not normallydistributed, the central limit theorem can beused (n > 25) … so the sampling distribution of isapproximately normal … with mean = 8 …and standard deviation(continued)xxμ0.5363nσσx ===
33. 33. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-33ExampleSolution (continued):(continued)0.38300.5)ZP(-0.53638-8.2nσμ-μ3638-7.8P8.2)μP(7.8 XX=<<=<<=<<Z7.8 8.2 -0.5 0.5SamplingDistributionStandard NormalDistribution .1915+.1915PopulationDistribution????????????Sample Standardize8μ = 8μX= 0μz =xX
34. 34. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-34Acceptance Intervals Goal: determine a range within which sample means arelikely to occur, given a population mean and variance By the Central Limit Theorem, we know that the distribution of Xis approximately normal if n is large enough, with mean μ andstandard deviation Let zα/2 be the z-value that leaves area α/2 in the upper tail of thenormal distribution (i.e., the interval - zα/2 to zα/2 encloses probability1 – α) Thenis the interval that includes X with probability 1 – αXσX/2σzμ α±
35. 35. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-35Sampling Distributions ofSample ProportionsSamplingDistributionsSamplingDistribution ofSampleMeanSamplingDistribution ofSampleProportionSamplingDistribution ofSampleVariance
36. 36. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-36Population Proportions, PP = the proportion of the population havingsome characteristic Sample proportion ( ) provides an estimateof P: 0 ≤ ≤ 1 has a binomial distribution, but can be approximatedby a normal distribution when nP(1 – P) > 9sizesampleinterestofsticcharacterithehavingsampletheinitemsofnumbernXP ==ˆPˆPˆPˆ
37. 37. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-37Sampling Distribution of P Normal approximation:Properties:and(where P = population proportion)Sampling Distribution.3.2.100 . 2 .4 .6 8 1p)PE( =ˆnP)P(1nXVarσ2P−==ˆ^)PP( ˆPˆ
38. 38. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-38Z-Value for ProportionsnP)P(1PPσPPZP−−=−=ˆˆˆStandardize to a Z value with the formula:Pˆ
39. 39. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-39Example If the true proportion of voters who supportProposition A is P = .4, what is the probabilitythat a sample of size 200 yields a sampleproportion between .40 and .45? i.e.: if P = .4 and n = 200, what isP(.40 ≤ ≤ .45) ?Pˆ
40. 40. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-40Example if P = .4 and n = 200, what isP(.40 ≤ ≤ .45) ?(continued).03464200.4).4(1nP)P(1σP=−=−=ˆ1.44)ZP(0.03464.40.45Z.03464.40.40P.45)PP(.40≤≤= −≤≤−=≤≤ ˆFind :Convert tostandardnormal:Pσ ˆPˆ
41. 41. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-41ExampleZ.45 1.44.4251StandardizeSampling DistributionStandardizedNormal Distribution if p = .4 and n = 200, what isP(.40 ≤ ≤ .45) ?(continued)Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251.40 0PˆPˆ
42. 42. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-42Sampling Distributions ofSample ProportionsSamplingDistributionsSamplingDistribution ofSampleMeanSamplingDistribution ofSampleProportionSamplingDistribution ofSampleVariance
43. 43. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-43Sample Variance Let x1, x2, . . . , xn be a random sample from apopulation. The sample variance is the square root of the sample variance is calledthe sample standard deviation the sample variance is different for differentrandom samples from the same population∑=−−=n1i2i2)x(x1n1s
44. 44. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-44Sampling Distribution ofSample Variances The sampling distribution of s2has mean σ2 If the population distribution is normal, then If the population distribution is normal thenhas a χ2distribution with n – 1 degrees of freedom22σ)E(s =1n2σ)Var(s42−=22σ1)s-(n
45. 45. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-45The Chi-square Distribution The chi-square distribution is a family of distributions,depending on degrees of freedom: d.f. = n – 1 Text Table 7 contains chi-square probabilities0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28 0 4 8 12 16 20 24 28d.f. = 1 d.f. = 5 d.f. = 15χ2χ2χ2
46. 46. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-46If the mean of these threevalues 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 varyfor a given mean)Idea: Number of observations that are free to varyafter sample mean has been calculatedExample: Suppose the mean of 3 numbers is 8.0Let X1 = 7Let X2 = 8What is X3?
47. 47. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-47 A commercial freezer must hold a selectedtemperature with little variation. Specifications callfor a standard deviation of no more than 4 degrees(a variance of 16 degrees2). A sample of 14 freezers is to betested What is the upper limit (K) for thesample variance such that theprobability of exceeding this limit,given that the population standarddeviation is 4, is less than 0.05?Chi-square Example
48. 48. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-48Finding the Chi-square Value Use the the chi-square distribution with area 0.05in the upper tail:probabilityα = .05χ213χ2χ213= 22.36= 22.36 (α = .05 and 14 – 1 = 13 d.f.)222σ1)s(n−=χIs chi-square distributed with (n – 1) = 13degrees of freedom
49. 49. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-49Chi-square Example0.05161)s(nPK)P(s 21322=>−=> χSo:(continued)χ213 = 22.36 (α = .05 and 14 – 1 = 13 d.f.)22.36161)K(n=−(where n = 14)so 27.521)(14)(22.36)(16K =−=If s2from the sample of size n = 14 is greater than 27.52, there isstrong evidence to suggest the population variance exceeds 16.or
50. 50. Statistics for Business andEconomics, 6e © 2007 PearsonEducation, Inc. Chap 7-50Chapter Summary Introduced sampling distributions Described the sampling distribution of sample means For normal populations Using the Central Limit Theorem Described the sampling distribution of sampleproportions Introduced the chi-square distribution Examined sampling distributions for sample variances Calculated probabilities using sampling distributions
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