Lecture 8: Sampling Methods
Lecture 8: Sampling MethodsDonglei Du(ddu@unb.edu)Faculty of Business Administration, University of New Brunswick, NB Canada FrederictonE3B 9Y2Donglei Du (UNB)ADM 2623: Business Statistics1 / 30
Table of contents1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics2 / 30
Layout1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics3 / 30
Why sampling?The physical impossibility of checking all items in the population,and, also, it would be too time-consumingThe studying of all the items in a population would not be costeffectiveThe sample results are usually adequateThe destructive nature of certain testsDonglei Du (UNB)ADM 2623: Business Statistics4 / 30
Sampling MethodsProbability Sampling: Each data unit in the population has a knownlikelihood of being included in the sample.Non-probability Sampling: Does not involve random selection;inclusion of an item is based on convenienceDonglei Du (UNB)ADM 2623: Business Statistics5 / 30
Sampling MethodsSampling with replacement: Each data unit in the population isallowed to appear in the sample more than once.Sampling without replacement: Each data unit in the population isallowed to appear in the sample no more than once.Donglei Du (UNB)ADM 2623: Business Statistics6 / 30
Random Sampling MethodsMost commonly used probability/random sampling techniques areSimple random samplingStratified random samplingCluster random samplingDonglei Du (UNB)ADM 2623: Business Statistics7 / 30
Simple random samplingEach item(person) in the population has an equal chance of beingincluded. Stratum 2Stratum 4Stratum 6IndexFigure: Credit: Open source textbook: OpenIntroStatistics, 2nd Edition, D. M.Diez, C. D. Barr, and M. Cetinkaya-RundelStratum 3(http://www.openintro.org/stat/textbook.php) ADM 2623: Business Statistics Donglei Du (UNB) Stratum 1 8 / 30
Stratified random sampling A population is first divided into strata which are made up of similarobservations. Take a simple random sample from each stratum. Stratum 4 Stratum 6 Stratum 3 Stratum 1 Index Stratum 2 Stratum 5 Cluster 9Cluster 5Cluster 2IndexCluster7Figure: Credit: Open source textbook: OpenIntroStatistics,2nd Edition, D. M.Diez, C. D. Barr, and M. Cetinkaya-RundelCluster 3(http://www.openintro.org/stat/textbook.php) ADM 2623:Business Statistics Cluster 8 Cluster 4 Donglei Du (UNB) 9 / 30
Stratum 3 Cluster random sampling 1A populationStratumis firstdivided into clusters which are usually not madeup of homogeneous observations, and take a simple random samplefrom a random sample of clusters. Stratum 5 Index Cluster 3 Cluster 6 Cluster 8 Cluster 4 Cluster 7 Cluster 9Cluster 5Cluster 2 Cluster 1Figure: Credit: Open source textbook: OpenIntro Statistics, 2nd Edition, D. M.Diez, C. D. Barr, and M. tbook.php)Donglei Du (UNB)ADM 2623: Business Statistics10 / 30
Layout1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics11 / 30
Simple random sampling without replacement (SRN)Repeat the following process until the requested sample is obtained:Randomly (with equal probability) select an item, record it, and discarditExample: draw cards one by one from a deck without replacement.This technique is the simplest and most often used samplingtechnique in practice.Donglei Du (UNB)ADM 2623: Business Statistics12 / 30
R codeGiven a population of size N , choose a sample of size n using SRN N -5 n -2 sample(1:N, n, replace FALSE)Donglei Du (UNB)ADM 2623: Business Statistics13 / 30
Simple random sampling with replacement (SRR)Repeat the following process until the requested sample is obtained:Randomly (with equal probability) select an item, record it, and replaceitExample: draw cards one by one from a deck with replacement.This is rarely used in practice, since there is no meaning to includethe same item more than once.However, it is preferred from a theoretical point of view, sinceIt is easy to analyze mathematically.Moreover, SRR is a very good approximation for SRN when N is large.Donglei Du (UNB)ADM 2623: Business Statistics14 / 30
R codeGiven a population {1, . . . , N } of size N , choose a sample of size nusing SRR N -5 n -2 sample(1:N, n, replace TRUE)Donglei Du (UNB)ADM 2623: Business Statistics15 / 30
Layout1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics16 / 30
Sampling error vs non-sampling errorSampling error: the difference between a sample statistic and itscorresponding population parameter. This error is inherent inThe sampling process (since sample is only part of the population)The choice of statistics (since a statistics is computed based on thesample).Non-sample Error: This error has no relationship to the samplingtechnique or the estimator. The main reasons are human-relateddata recordingnon-responsesample selectionDonglei Du (UNB)ADM 2623: Business Statistics17 / 30
Layout1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics18 / 30
Sampling distribution of sample statisticSampling distribution of sample statistic: The probability distributionconsisting of all possible sample statistics of a given sample sizeselected from a population using one probability sampling.Example: we can consider the sampling distribution of the samplemean, sample variance etc.Donglei Du (UNB)ADM 2623: Business Statistics19 / 30
An example of the sampling distribution of sample meanunder SRRConsider a small population {1, 2, 3, 4, 5} with size N 5. Let usrandomly choose a sample of size n 2 via SRR.It is understood that sample is ordered. Then there areN n 52 25 possible samples; onglei Du sample(3,1)(3,2)(3,3)(3,4)(3,5)x̄22.533.54ADM 2623: Business 544.5sample(5,1)(5,2)(5,1)(5,1)(5,1)x̄33.544.5520 / 30
An example of the sampling distribution of sample meanunder SRRLet us find the sampling distribution of the sample mean:X̄11.522.533.544.55Donglei Du 25ADM 2623: Business Statistics21 / 30
The mean and variance of the sample mean under SRRLet us find the mean and variance of the sampling distribution of thesample mean:X̄11.522.533.544.55Donglei Du (UNB)P (X̄)1/252/253/254/255/254/253/252/251/25X̄P (X̄)1/253/256/2510/2515/2514/2512/259/255/2575/25 3ADM 2623: Business StatisticsX̄ 2 P 25250/25 1022 / 30
The mean and variance of the sample mean under SRRSo the mean and variance of the sample mean are given asx̄ 3s2 10 32 1On the other hand, the population mean and variance are given asµ σ2 Donglei Du (UNB)1 2. 5 35255 155 25ADM 2623: Business Statistics23 / 30
Relationship between sample and population mean andvariance under SRRSo from this examplex̄ µ 32σ2 1s2 22The above relationship is true for any population of size N andsample of size nx̄ µσ2s2 nDonglei Du (UNB)ADM 2623: Business Statistics24 / 30
Distribution of the sample mean under SRRHistogram of x012Frequency345Let us look the histogram of the sample mean in the above example.12345xDonglei Du (UNB)ADM 2623: Business Statistics25 / 30
Distribution of the sample mean under SRR for variouspopulationLet us look the histogram of the sample mean for various population.Donglei Du (UNB)ADM 2623: Business Statistics26 / 30
Layout1Sampling MethodsWhy SamplingProbability vs non-probability sampling methodsSampling with replacement vs without replacementRandom Sampling Methods2Simple random sampling with and without replacementSimple random sampling without replacementSimple random sampling with replacement3Sampling error vs non-sampling error4Sampling distribution of sample statisticHistogram of the sample mean under SRR5Distribution of the sample mean under SRR: The central limit theoremDonglei Du (UNB)ADM 2623: Business Statistics27 / 30
Distribution of the sample mean under SRR: The centrallimit theoremThe central limit theorem: The sampling distribution of the meansof all possible samples of size n generated from the population usingSRR will be approximately normally distributed when n goes toinfinity.X̄ µ N (0, 1)σ/ nHow large should n be for the sampling mean distribution to beapproximately normal?In practice, n 30If n large, and we do not know σ, then we can use sample standarddeviation instead. Then Central Limit Theorem is still true!Donglei Du (UNB)ADM 2623: Business Statistics28 / 30
Distribution of the sample mean under SRR for smallsampleIf n small, and we do not know σ, but we know the population isnormally distributed, then replacing the standard deviation withsample standard deviation results in the Student’s t distribution withdegrees of freedom df n 1:T X̄ µ t(n 1)s/ nLike Z, the t-distribution is continuousTakes values between and It is bell-shaped and symmetric about zeroIt is more spread out and flatter at the center than the z-distributionFor larger and larger values of degrees of freedom, the t-distributionbecomes closer and closer to the standard normal distributionDonglei Du (UNB)ADM 2623: Business Statistics29 / 30
Comparison of t Distributions with Normal distribution0.4Comparison of t DistributionsDistributions0.20.00.1Density0.3df 1df 3df 8df 30normal 4 2024x valueDonglei Du (UNB)ADM 2623: Business Statistics30 / 30
Lecture 8: Sampling Methods Donglei Du (ddu@unb.edu) . The sampling process (since sample is only part of the population) The choice of statistics (since a statistics is computed based on the sample). . 1 Sampling Methods Why Sampling Probability vs non-probability sampling methods
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