Chapter 8 Sampling Distributions

Chapter 8

Sampling Distributions

by Try Sothearith [email protected] [email protected] Tel: 012 585 865 / 016555507

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.

Learning ObjectivesIn this chapter, you learn: The concept of the sampling distributionTo compute probabilities related to the sample mean and the sample proportionThe importance of the Central Limit TheoremTo distinguish between different survey sampling methodsTo evaluate survey worthiness and survey errors

Sampling DistributionsSampling DistributionsSampling Distribution of the MeanSampling Distribution of the Proportion

Sampling DistributionsA sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population

Developing a Sampling DistributionAssume there is a population Population size N=4Random variable, X, is age of individualsValues of X: 18, 20, 22, 24 (years)ABCD

.3.2.1 0 18 20 22 24 A B C DUniform DistributionP(x)x(continued)Summary Measures for the Population Distribution:Developing a Sampling Distribution

16 possible samples (sampling with replacement)Now consider all possible samples of size n=2(continued)Developing a Sampling Distribution16 Sample Means

1stObs2nd Observation182022241818,1818,2018,2218,242020,1820,2020,2220,242222,1822,2022,2222,242424,1824,2024,2224,24

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Sampling Distribution of All Sample Means18 19 20 21 22 23 240 .1 .2 .3 P(X) XSample Means Distribution16 Sample Means_Developing a Sampling Distribution(continued)(no longer uniform)_

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Summary Measures of this Sampling Distribution:Developing aSampling Distribution(continued)

Comparing the Population with its Sampling Distribution18 19 20 21 22 23 240 .1 .2 .3 P(X) X 18 20 22 24 A B C D0 .1 .2 .3 PopulationN = 4P(X) X_Sample Means Distributionn = 2_

Sampling Distribution of the MeanSampling DistributionsSampling Distribution of the MeanSampling Distribution of the Proportion

Standard Error of the MeanDifferent samples of the same size from the same population will yield different sample meansA measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

Note that the standard error of the mean decreases as the sample size increases

If the Population is NormalIf a population is normal with mean and standard deviation , the sampling distribution of is also normally distributed with

and

Z-value for Sampling Distributionof the MeanZ-value for the sampling distribution of :where:= sample mean= population mean= population standard deviation n = sample size

Normal Population DistributionNormal Sampling Distribution (has the same mean)Sampling Distribution Properties

(i.e. is unbiased )

Sampling Distribution Properties

As n increases, decreasesLarger sample sizeSmaller sample size(continued)

If the Population is not NormalWe can apply the Central Limit Theorem:Even if the population is not normal,sample means from the population will be approximately normal as long as the sample size is large enough.

Properties of the sampling distribution:

and

nCentral Limit TheoremAs the sample size gets large enough the sampling distribution becomes almost normal regardless of shape of population

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 7-*Population DistributionSampling Distribution (becomes normal as n increases)Central TendencyVariationLarger sample sizeSmaller sample sizeIf the Population is not Normal(continued)Sampling distribution properties:


How Large is Large Enough?For most distributions, n > 30 will give a sampling distribution that is nearly normalFor fairly symmetric distributions, n > 15For normal population distributions, the sampling distribution of the mean is always normally distributed

ExampleSuppose a population has mean = 8 and standard deviation = 3. Suppose a random sample of size n = 36 is selected.

What is the probability that the sample mean is between 7.8 and 8.2?

ExampleSolution:Even if the population is not normally distributed, the central limit theorem can be used (n > 30) so the sampling distribution of is approximately normal with mean = 8 and standard deviation

(continued)

Example Solution (continued):

(continued)Z7.8 8.2-0.4 0.4Sampling DistributionStandard Normal Distribution.1554 +.1554Population Distribution????????????SampleStandardizeX

Sampling Distribution of the ProportionSampling DistributionsSampling Distribution of the MeanSampling Distribution of the Proportion

Population Proportions = CasmamaRtrbs;saklsmamaRtKMrU ( p ) pl;nYvkar):an;sanmamaRtrbs;sakl :

0 p 1X cMnYnkrNIekIteLIgn cMnYnkrNIsrub

Sampling Distribution of pRKb;r)ayEdlman

Car)aynrm:al

where and(where = population proportion)Sampling DistributionP( ps).3.2.1 0 0 . 2 .4 .6 8 1p

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 7-*Z-Value for ProportionsStandardize p to a Z value with the formula:


Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 7-*Example]bmafakare)aHeqatKMRTman = 0.4 KNnaRbU)ablIetntMl nsmamaRtBI 0.40 nig 0.45 ebIKMrUmanTMhM 200?i.e.: if = 0.4 and n = 200, what is P(0.40 p 0.45) ?


Example if = 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)Find : Convert to standard normal:

Basic Business Statistics, 10e 2006 Prentice-Hall, Inc.Chap 7-*ExampleZ0.451.440.4251StandardizeSampling DistributionStandardized Normal Distribution if = 0.4 and n = 200, what is P(0.40 p 0.45) ?(continued)Use standard normal table: P(0 Z 1.44) = 0.42510.400p


Types of Samples UsedQuotaSamplesNon-Probability SamplesJudgementProbability SamplesSimple RandomSystematicStratifiedClusterConvenience(continued)

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Chapter 8 Sampling Distributions