Chap 20-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 20 Sampling: Additional Topics in Sampling Statistics for Business.

Chap 20-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Chapter 20

Sampling:Additional Topics in Sampling

Statistics for Business and Economics

6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-2

Chapter Goals

After completing this chapter, you should be able to:

Explain the basic steps of a sampling study Describe sampling and nonsampling errors Explain simple random sampling and stratified sampling Analyze results from simple random or stratified

samples Determine sample size when estimating population

mean, population total, or population proportion Describe other sampling methods

Cluster Sampling, Two-Phase Sampling, Nonprobability Samples


Steps of a Sampling Study

Step 1: Information Required?

Step 2: Relevant Population?

Step 3: Sample Selection?

Step 4: Obtaining Information?

Step 5: Inferences From

Step 6: Conclusions?


Sampling and Nonsampling Errors

A sample statistic is an estimate of an unknown population parameter

Sample evidence from a population is variable Sample-to-sample variation is expected

Sampling error results from the fact that we only see a subset of the population when a sample is selected

Statistical statements can be made about sampling error It can be measured and interpreted using confidence

intervals, probabilities, etc.


Sampling and Nonsampling Errors

Nonsampling error results from sources not related to the sampling procedure used

Examples: The population actually sampled is not the relevant

one Survey subjects may give inaccurate or dishonest

answers Nonresponse to survey questions

(continued)


Probability Sample Items in the sample are chosen on the

basis of known probabilities

Nonprobability Sample Items included are chosen without

regard to their probability of occurrence

Types of Samples


Types of Samples

Quota

Samples

Non-Probability Samples

Judgement Convenience

(continued)

Probability Samples

Simple Random

Systematic

Stratified

Cluster


Simple Random Samples

Suppose that a sample of n objects is to be selected from a population of N objects

A simple random sample procedure is one in which every possible sample of n objects is equally likely to be chosen

Only sampling without replacement is considered here Random samples can be obtained from table of random

numbers or computer random number generators


Decide on sample size: n Divide frame of N individuals into groups of j

individuals: j=N/n Randomly select one individual from the 1st

group Select every jth individual thereafter

Systematic Sampling

N = 64

n = 8

j = 8

First Group


Finite Population Correction Factor

Suppose sampling is without replacement and the sample size is large relative to the population size

Assume the population size is large enough to apply the central limit theorem

Apply the finite population correction factor when estimating the population variance

N

nNfactor correction population finite


Estimating the Population Mean

Let a simple random sample of size n be taken from a population of N members with mean μ

The sample mean is an unbiased estimator of the population mean μ

The point estimate is:

n

1iix

n

1x


Estimating the Population Mean

An unbiased estimation procedure for the variance of the sample mean yields the point estimate

Provided the sample size is large, 100(1 - )% confidence intervals for the population mean are given by

N

nN

n

sσ

22x

ˆ

xα/2xα/2 σzxμσzx ˆˆ

(continued)


Estimating the Population Total

Consider a simple random sample of size n from a population of size N

The quantity to be estimated is the population total Nμ

An unbiased estimation procedure for the population total Nμ yields the point estimate NX


Estimating the Population Total

An unbiased estimator of the variance of the population total is

Provided the sample size is large, a 100(1 - )% confidence interval for the population total is

xα/2xα/2 σNzxNNμσNzxN ˆˆ

n)N(Nn

sσN

22x

2 ˆ


Confidence Interval for Population Total: Example

A firm has a population of 1000 accounts and wishes to estimate the total population value

A sample of 80 accounts is selected with average balance of $87.6 and standard deviation of $22.3

Find the 95% confidence interval estimate of the total balance


Example Solution

The 95% confidence interval for the population total balance is $82,912.52 to $92,287.16

2391.415718835σN

5718835)(1000)(92080

(22.3)n)N(N

n

sσN

x

222x

2

ˆ

ˆ

22.3s 87.6,x 80, n 1000,N

1.41)(1.96)(2396)(1000)(87.σNzxN xα/2 ˆ

92287.16Nμ82912.84


Estimating the Population Proportion

Let the true population proportion be P

Let be the sample proportion from n observations from a simple random sample

The sample proportion, , is an unbiased estimator of the population proportion, P

p̂

p̂


Estimating the Population Proportion

An unbiased estimator for the variance of the population proportion is

Provided the sample size is large, a 100(1 - )% confidence interval for the population proportion is

pα/2pα/2 σzpPσzp ˆˆˆˆˆˆ

N

n)(N

1n

)p(1pσ2

p

ˆˆ

ˆˆ

(continued)


Stratified Sampling

Overview of stratified sampling:

Divide population into two or more subgroups (called

strata) according to some common characteristic

A simple random sample is selected from each subgroup

Samples from subgroups are combined into one

Population

Divided

into 4

strata

Sample


Stratified Random Sampling

Suppose that a population of N individuals can be subdivided into K mutually exclusive and collectively exhaustive groups, or strata

Stratified random sampling is the selection of independent simple random samples from each stratum of the population.

Let the K strata in the population contain N1, N2,. . ., NK members, so that N1 + N2 + . . . + NK = N

Let the numbers in the samples be n1, n2, . . ., nK. Then the total number of sample members is

n1 + n2 + . . . + nK = n


Estimation of the Population Mean, Stratified Random Sample

Let random samples of nj individuals be taken from strata containing Nj individuals (j = 1, 2, . . ., K)

Let

Denote the sample means and variances in the strata by Xj and sj

2 and the overall population mean by μ

An unbiased estimator of the overall population mean μ is:

K

1jjjst xN

N

1x

K

1j

K

1jjj nnandNN


Estimation of the Population Mean, Stratified Random Sample

An unbiased estimator for the variance of the overall population mean is

where

Provided the sample size is large, a 100(1 - )% confidence interval for the population mean for stratified random samples is

(continued)

2x

K

1j

2j2

2x jst

σNN

1σ ˆˆ

stst xα/2stxα/2st σzxμσzx ˆˆ

j

jj

j

2j2

x N

)n(N

n

sσ

j

ˆ


Estimation of the Population Total, Stratified Random Sample

Suppose that random samples of nj individuals from strata containing Nj individuals (j = 1, 2, . . ., K) are selected and that the quantity to be estimated is the population total, Nμ

An unbiased estimation procedure for the population total Nμ yields the point estimate

K

1jjjst xNxN


Estimation of the Population Total, Stratified Random Sample

An unbiased estimation procedure for the variance of the estimator of the population total yields the point estimate

Provided the sample size is large, 100(1 - )% confidence intervals for the population total for stratified random samples are obtained from

(continued)

stα/2ststα/2st σNzxNNμσNzxN ˆˆ

2x

K

1j

2j

2x

2

ststσNσN ˆˆ


Estimation of the Population Proportion, Stratified Random Sample

Suppose that random samples of nj individuals from strata containing Nj individuals (j = 1, 2, . . ., K) are obtained

Let Pj be the population proportion, and the sample proportion, in the jth stratum

If P is the overall population proportion, an unbiased estimation procedure for P yields

K

1jjjst pN

N

1p ˆˆ

jp̂


• An unbiased estimation procedure for the variance of the estimator of the overall population proportion is

where

is the estimate of the variance of the sample proportion in the jth stratum

(continued)


2p

K

1j

2j2

2p jst

σNN

1σ ˆˆ

ˆˆ

j

jj

j

jj2p N

)n(N

1n

)p(1pσ

j

ˆˆ

ˆˆ


Provided the sample size is large, 100(1 - )% confidence intervals for the population proportion for stratified random samples are obtained from

(continued)


stst pα/2stpα/2st σzpPσzp ˆˆˆˆˆˆ


Proportional Allocation: Sample Size

One way to allocate sampling effort is to make the proportion of sample members in any stratum the same as the proportion of population members in the stratum

If so, for the jth stratum,

The sample size for the jth stratum using proportional allocation is

N

N

n

n jj

nN

Nn j

j


Optimal Allocation

To estimate an overall population mean or total and if the population variances in the individual strata are denoted σj

2 , the most precise estimators are obtained with optimal allocation

The sample size for the jth stratum using optimal allocation is

nσN

σNn K

1iii

jjj


Optimal Allocation

To estimate the overall population proportion, estimators with the smallest possible variance are obtained by optimal allocation

The sample size for the jth stratum for population proportion using optimal allocation is

(continued)

n)P(1PN

)P(1PNn K

1iiii

jjj

j


Determining Sample Size

The sample size is directly related to the size of the variance of the population estimator

If the researcher sets the allowable size of the variance in advance, the necessary sample size can be determined


Sample Size, Mean, Simple Random Sampling

Consider estimating the mean of a population of N members, which has variance σ2

If the desired variance, of the sample mean is specified, the required sample size to estimate the population mean through simple random sampling is

22x

2

σ1)σ(N

Nσn

2xσ


Sample Size, Mean, Simple Random Sampling

Often it is more convenient to specify directly the desired width of the confidence interval for the population mean rather than

Thus the researcher specifies the desired margin of error for the mean

Calculations are simple since, for example, a 95% confidence interval for the population mean will extend an approximate amount 1.96 on each side of the sample mean, X

2xσ

(continued)

xσ


Required Sample Size Example

2000 items are in a population. If σ = 45, what sample size is needed to estimate the mean within ± 5 with 95% confidence?

(Always round up)

So the required sample size is n = 270

269.39(45)51)(1999)(2.5

(2000)(45)

σ1)σ(N

Nσn

22

2

22x

2

N = 2000, 1.96 = 5 → = 2.551 xσ xσ


Consider estimating the proportion P of individuals in a population of size N who possess a certain attribute

If the desired variance, , of the sample proportion is specified, the required sample size to estimate the population proportion through simple random sampling is

Sample Size, Proportion,Simple Random Sampling

(continued)

P)P(11)σ(N

P)NP(1n

2p

ˆ

2pσ ˆ


The largest possible value for this expression occurs when the value of P is 0.25

A 95% confidence interval for the population proportion will extend an approximate amount 1.96 on each side of the sample proportion

Sample Size, Proportion,Simple Random Sampling

(continued)

0.251)σ(N

0.25Nn

2p

max

ˆ

pσ ˆ



How large a sample would be necessary to estimate the true proportion of voters who will vote for proposition A, within ±3%, with 95% confidence, from a population of 3400 voters?



Solution:

N = 34000

For 95% confidence, use z = 1.96

1.96 = .03 → = .015306

So use n = 1036

(continued)

spσ ˆ spσ ˆ

1035.47025153)(33999)(.0

00)(0.25)(340

0.251)σ(N

0.25Nn

22p

max

ˆ


Sample Size, Mean, Stratified Sampling

Suppose that a population of N members is subdivided in K strata containing N1, N2, . . .,NK members

Let σj2 denote the population variance in the jth stratum

An estimate of the overall population mean is desired

If the desired variance, , of the sample estimator is specified, the required total sample size, n, can be found

2xst

σ


Sample Size, Mean, Stratified Sampling

For proportional allocation:

For optimal allocation:

K

1j

2jj

2x

K

1j

2jj

σNN1

Nσ

σN

n

st

(continued)

K

1j

2jj

2x

K

1j

2jj

σNN1

Nσ

σNN1

n

st


Cluster Sampling

Population is divided into several “clusters,” each representative of the population

A simple random sample of clusters is selected Generally, all items in the selected clusters are examined An alternative is to chose items from selected clusters using

another probability sampling technique

Population divided into 16 clusters. Randomly selected

clusters for sample


Estimators for Cluster Sampling

A population is subdivided into M clusters and a simple random sample of m of these clusters is selected and information is obtained from every member of the sampled clusters

Let n1, n2, . . ., nm denote the numbers of members in

the m sampled clusters

Denote the means of these clusters by

Denote the proportions of cluster members possessing an attribute of interest by P1, P2, . . . , Pm

m21 x,,x,x


Estimators for Cluster Sampling

The objective is to estimate the overall population mean µ and proportion P

Unbiased estimation procedures give

Mean Proportion

m

1ii

m

1iii

c

n

xnx

m

1ii

m

1iii

c

n

pnp̂

(continued)


Estimates of the variance of these estimators, following from unbiased estimation procedures, are

Mean Proportion

1m

)xx(n

nMm

mMσ

m

1i

2ci

2i

22xc

ˆ

1m

)p(Pn

nMm

mMσ

m

1i

2ci

2i

22pc

ˆˆ

ˆ

m

nn

m

1ii

Where is the average number of individuals in the sampled clusters

Estimators for Cluster Sampling(continued)


Provided the sample size is large, 100(1 - )% confidence intervals using cluster sampling are

for the population mean

for the population proportion

cc xα/2cxα/2c σzxμσzx ˆˆ

cc pα/2cpα/2c σzpPσzp ˆˆˆˆˆˆ

Estimators for Cluster Sampling(continued)


Two-Phase Sampling

Sometimes sampling is done in two steps An initial pilot sample can be done Disadvantage:

takes more time Advantages:

Can adjust survey questions if problems are noted Additional questions may be identified Initial estimates of response rate or population

parameters can be obtained



Quota

Samples


Judgement Convenience

Probability Samples

Simple Random

Systematic

Stratified

Cluster



It may be simpler or less costly to use a non-probability based sampling method Judgement sample Quota sample Convience sample

These methods may still produce good estimates of population parameters

But … Are more subject to bias No valid way to determine reliability

(continued)


Chapter Summary

Reviewed basic steps in a sampling study Defined sampling and nonsampling errors Examined probability sampling methods

Simple Random Sampling, Systematic Sampling, Stratified Random Sampling, Cluster Sampling

Identified Estimators for the population mean, population total, and population proportion for different types of samples

Determined the required sample size for specified confidence interval width

Examined nonprobabilistic sampling methods

Chap 20-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 20 Sampling: Additional Topics in Sampling Statistics for Business.

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relevant population

population varianceestimating

population proportiondescribe

population totalconsider

population meanlet

population sizeassume

sample mean

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