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### Ch07

1. 1. Copyright ©2012 Pearson Education Chap 7-1Chap 7-1Chapter 7Sampling and Sampling DistributionsBasic Business Statistics for12th Edition
2. 2. Copyright ©2012 Pearson Education Chap 7-2Copyright ©2012 Pearson Education Chap 7-2Learning ObjectivesIn this chapter, you learn: To distinguish between different samplingmethods The concept of the sampling distribution To compute probabilities related to the samplemean and the sample proportion The importance of the Central Limit Theorem
3. 3. Copyright ©2012 Pearson Education Chap 7-3Copyright ©2012 Pearson Education Chap 7-3Why Sample? Selecting a sample is less time-consuming& less costly than selecting every item inthe population (census). An analysis of a sample is lesscumbersome and more practical than ananalysis of the entire population.DCOVA
4. 4. Copyright ©2012 Pearson Education Chap 7-4Copyright ©2012 Pearson Education Chap 7-4A Sampling Process Begins With ASampling Frame The sampling frame is a listing of items thatmake up the population Frames are data sources such as populationlists, directories, or maps Inaccurate or biased results can result if aframe excludes certain portions of thepopulation Using different frames to generate data canlead to dissimilar conclusionsDCOVA
6. 6. Copyright ©2012 Pearson Education Chap 7-6Copyright ©2012 Pearson Education Chap 7-6Types of Samples:Nonprobability Sample In a nonprobability sample, items included arechosen without regard to their probability ofoccurrence. In convenience sampling, items are selected basedonly on the fact that they are easy, inexpensive, orconvenient to sample. In a judgment sample, you get the opinions of pre-selected experts in the subject matter.DCOVA
7. 7. Copyright ©2012 Pearson Education Chap 7-7Copyright ©2012 Pearson Education Chap 7-7Types of Samples:Probability Sample In a probability sample, items in thesample are chosen on the basis of knownprobabilities.Probability SamplesSimpleRandomSystematic Stratified ClusterDCOVA
8. 8. Copyright ©2012 Pearson Education Chap 7-8Copyright ©2012 Pearson Education Chap 7-8Probability Sample:Simple Random Sample Every individual or item from the frame has anequal chance of being selected. Selection may be with replacement (selectedindividual is returned to frame for possiblereselection) or without replacement (selectedindividual isn’t returned to the frame). Samples obtained from table of randomnumbers or computer random numbergenerators.DCOVA
9. 9. Copyright ©2012 Pearson Education Chap 7-9Copyright ©2012 Pearson Education Chap 7-9Selecting a Simple Random SampleUsing A Random Number TableSampling Frame ForPopulation With 850ItemsItem Name Item #Bev R. 001Ulan X. 002. .. .. .. .Joann P. 849Paul F. 850Portion Of A Random Number Table49280 88924 35779 00283 81163 0727511100 02340 12860 74697 96644 8943909893 23997 20048 49420 88872 08401The First 5 Items in a simplerandom sampleItem # 492Item # 808Item # 892 -- does not exist so ignoreItem # 435Item # 779Item # 002DCOVA
10. 10. Copyright ©2012 Pearson Education Chap 7-10Copyright ©2012 Pearson Education Chap 7-10 Decide on sample size: n Divide frame of N individuals into groups of kindividuals: k=N/n Randomly select one individual from the 1stgroup Select every kth individual thereafterProbability Sample:Systematic SampleN = 40n = 4k = 10First GroupDCOVA
11. 11. Copyright ©2012 Pearson Education Chap 7-11Copyright ©2012 Pearson Education Chap 7-11Probability Sample:Stratified Sample Divide population into two or more subgroups (called strata) accordingto some common characteristic A simple random sample is selected from each subgroup, with samplesizes proportional to strata sizes Samples from subgroups are combined into one This is a common technique when sampling population of voters,stratifying across racial or socio-economic lines.PopulationDividedinto 4strataDCOVA
12. 12. Copyright ©2012 Pearson Education Chap 7-12Copyright ©2012 Pearson Education Chap 7-12Probability SampleCluster Sample Population is divided into several “clusters,” each representative ofthe population A simple random sample of clusters is selected All items in the selected clusters can be used, or items can bechosen from a cluster using another probability sampling technique A common application of cluster sampling involves election exit polls,where certain election districts are selected and sampled.Populationdivided into16 clusters. Randomly selectedclusters for sampleDCOVA
13. 13. Copyright ©2012 Pearson Education Chap 7-13Copyright ©2012 Pearson Education Chap 7-13Probability Sample:Comparing Sampling Methods Simple random sample and Systematic sample Simple to use May not be a good representation of the population’sunderlying characteristics Stratified sample Ensures representation of individuals across the entirepopulation Cluster sample More cost effective Less efficient (need larger sample to acquire the samelevel of precision)DCOVA
14. 14. Copyright ©2012 Pearson Education Chap 7-14Copyright ©2012 Pearson Education Chap 7-14Evaluating Survey Worthiness What is the purpose of the survey? Is the survey based on a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit goodresponses Sampling error – always existsDCOVA
15. 15. Copyright ©2012 Pearson Education Chap 7-15Copyright ©2012 Pearson Education Chap 7-15Types of Survey Errors Coverage error or selection bias Exists if some groups are excluded from the frame and haveno chance of being selected Nonresponse error or bias People who do not respond may be different from those whodo respond Sampling error Variation from sample to sample will always exist Measurement error Due to weaknesses in question design, respondent error, andinterviewer’s effects on the respondent (“Hawthorne effect”)DCOVA
17. 17. Copyright ©2012 Pearson Education Chap 7-17Copyright ©2012 Pearson Education Chap 7-17Sampling Distributions A sampling distribution is a distribution of all of thepossible values of a sample statistic for a given sizesample selected from a population. For example, suppose you sample 50 students from yourcollege regarding their mean GPA. If you obtained manydifferent samples of 50, you will compute a differentmean for each sample. We are interested in thedistribution of the mean GPA from all possible samples of50 students.DCOVA
18. 18. Copyright ©2012 Pearson Education Chap 7-18Copyright ©2012 Pearson Education Chap 7-18Developing 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 DDCOVA
19. 19. Copyright ©2012 Pearson Education Chap 7-19Copyright ©2012 Pearson Education Chap 7-19.3.2.1018 20 22 24A B C DUniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing aSampling Distribution21424222018NXμ i2.236Nμ)(Xσ2iDCOVA
20. 20. Copyright ©2012 Pearson Education Chap 7-20Copyright ©2012 Pearson Education Chap 7-2016 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 SampleMeans1stObs2nd Observation18 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 24,24DCOVA
21. 21. Copyright ©2012 Pearson Education Chap 7-21Copyright ©2012 Pearson Education Chap 7-211st 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)_DCOVA
22. 22. Copyright ©2012 Pearson Education Chap 7-22Copyright ©2012 Pearson Education Chap 7-22Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)211624191918μX1.581621)-(2421)-(1921)-(18σ222XDCOVANote: Here we divide by 16 because there are 16different samples of size 2.
23. 23. Copyright ©2012 Pearson Education Chap 7-23Copyright ©2012 Pearson Education Chap 7-23Comparing the Population Distributionto the Sample Means 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 XX2.236σ21μSample Means Distributionn = 2_DCOVA
24. 24. Copyright ©2012 Pearson Education Chap 7-24Copyright ©2012 Pearson Education Chap 7-24Sampling Distribution of The Mean:Standard 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:(This assumes that sampling is with replacement orsampling is without replacement from an infinite population) Note that the standard error of the mean decreases asthe sample size increasesnσσXDCOVA
25. 25. Copyright ©2012 Pearson Education Chap 7-25Copyright ©2012 Pearson Education Chap 7-25Sampling Distribution of The Mean:If the Population is Normal If a population is normal with mean μ andstandard deviation σ, the sampling distributionof is also normally distributed withandXμμXnσσXDCOVA
26. 26. Copyright ©2012 Pearson Education Chap 7-26Copyright ©2012 Pearson Education Chap 7-26Z-value for Sampling Distributionof the Mean Z-value for the sampling distribution of :where: = sample mean= population mean= population standard deviationn = sample sizeXμσnσμ)X(σ)μX(ZXXXDCOVA
29. 29. Copyright ©2012 Pearson Education Chap 7-29Copyright ©2012 Pearson Education Chap 7-29Determining An Interval Including AFixed Proportion of the Sample MeansFind a symmetrically distributed interval around µthat will include 95% of the sample means when µ= 368, σ = 15, and n = 25. Since the interval contains 95% of the sample means5% of the sample means will be outside the interval. Since the interval is symmetric 2.5% will be abovethe upper limit and 2.5% will be below the lower limit. From the standardized normal table, the Z score with2.5% (0.0250) below it is -1.96 and the Z score with2.5% (0.0250) above it is 1.96.DCOVA
30. 30. Copyright ©2012 Pearson Education Chap 7-30Copyright ©2012 Pearson Education Chap 7-30Determining An Interval Including AFixed Proportion of the Sample Means Calculating the lower limit of the interval Calculating the upper limit of the interval 95% of all sample means of sample size 25 arebetween 362.12 and 373.8812.3622515)96.1(368nZX Lσμ(continued)88.3732515)96.1(368nZXUσμDCOVA
31. 31. Copyright ©2012 Pearson Education Chap 7-31Copyright ©2012 Pearson Education Chap 7-31Sampling Distribution of The Mean:If 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:andμμxnσσxDCOVA
32. 32. Copyright ©2012 Pearson Education Chap 7-32Chap 7-32Population DistributionSampling Distribution(becomes normal as n increases)Central TendencyVariationxxLargersamplesizeSmallersample sizeSample Mean Sampling Distribution:If the Population is not Normal(continued)Sampling distributionproperties:μμxnσσxxμμDCOVA
33. 33. Copyright ©2012 Pearson Education Chap 7-33Copyright ©2012 Pearson Education Chap 7-33How Large is Large Enough? For most distributions, n ≥ 30 will give asampling distribution that is nearly normal For fairly symmetric distributions, n ≥ 15 For normal population distributions, thesampling distribution of the mean is alwaysnormally distributedDCOVA
34. 34. Copyright ©2012 Pearson Education Chap 7-34Copyright ©2012 Pearson Education Chap 7-34Example 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?DCOVA
35. 35. Copyright ©2012 Pearson Education Chap 7-35Copyright ©2012 Pearson Education Chap 7-35ExampleSolution: Even if the population is not normallydistributed, the central limit theorem can beused (n ≥ 30) … so the sampling distribution of isapproximately normal … with mean = 8 …and standard deviation(continued)xxμ0.5363nσσxDCOVA
37. 37. Copyright ©2012 Pearson Education Chap 7-37Copyright ©2012 Pearson Education Chap 7-37Population Proportionsπ = the proportion of the population havinga characteristic of interest Sample proportion (p) provides an estimateof π: 0 ≤ p ≤ 1 p is approximately distributed as a normal distributionwhen n is large(assuming sampling with replacement from a finite population orwithout replacement from an infinite population)sizesampleinterestofsticcharacterithehavingsampletheinitemsofnumbernXpDCOVA
38. 38. Copyright ©2012 Pearson Education Chap 7-38Copyright ©2012 Pearson Education Chap 7-38Sampling Distribution of p Approximated by anormal distribution if:whereand(where π = population proportion)Sampling DistributionP(ps).3.2.100 . 2 .4 .6 8 1 pπpμn)(1σpππ5)n(15nandππDCOVA
39. 39. Copyright ©2012 Pearson Education Chap 7-39Copyright ©2012 Pearson Education Chap 7-39Z-Value for Proportionsn)(1pσpZpStandardize p to a Z value with the formula:DCOVA
40. 40. Copyright ©2012 Pearson Education Chap 7-40Copyright ©2012 Pearson Education Chap 7-40Example If the true proportion of voters who supportProposition A is π = 0.4, what is the probabilitythat a sample of size 200 yields a sampleproportion between 0.40 and 0.45? i.e.: if π = 0.4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?DCOVA
41. 41. Copyright ©2012 Pearson Education Chap 7-41Copyright ©2012 Pearson Education Chap 7-41Example if π = 0.4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?(continued)0.034642000.4)0.4(1n)(1σp1.44)ZP(00.034640.400.45Z0.034640.400.40P0.45)pP(0.40Find :Convert tostandardizednormal:pσDCOVA
42. 42. Copyright ©2012 Pearson Education Chap 7-42Copyright ©2012 Pearson Education Chap 7-42ExampleZ0.45 1.440.4251StandardizeSampling DistributionStandardizedNormal Distribution if π = 0.4 and n = 200, what isP(0.40 ≤ p ≤ 0.45) ?(continued)Utilize the cumulative normal table:P(0 ≤ Z ≤ 1.44) = 0.9251 – 0.5000 = 0.42510.40 0pDCOVA
43. 43. Copyright ©2012 Pearson Education Chap 7-43Copyright ©2012 Pearson Education Chap 7-43Chapter Summary Discussed probability and nonprobability samples Described four common probability samples Examined survey worthiness and types of surveyerrors Introduced sampling distributions Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion Calculated probabilities using sampling distributions