Sampling and Sampling Distributions

Dr. Himani Gupta

Why Sample? Selecting a sample is less time-consuming than selecting

every item in the population (census). Selecting a sample is less costly than selecting every item

in the population. An analysis of a sample is less cumbersome and more

practical than an analysis of the entire population.For example: Sampling (i.e. selecting a sub-set of a whole

population) is often done for reasons of cost (it’s less expensive to sample 1,000 television viewers than 100 million TV viewers) and practicality (e.g. performing a crash test on every automobile produced is impractical).

Dr. Himani Gupta

Types of Samples

Quota

Samples

Non-Probability Samples

Judgment snowball

Probability Samples

Simple Random

Systematic

Stratified

ClusterConvenience

Dr. Himani Gupta

Nonrandom Sampling Convenience Sampling: sample elements are selected for

the convenience of the researcher

Judgment Sampling: sample elements are selected by the judgment of the researcher. you get the opinions of pre-selected experts in the subject matter

Quota Sampling: sample elements are selected until the quota controls are satisfied

Snowball Sampling: survey subjects are selected based on referral from other survey respondents

Dr. Himani Gupta

Convenience Sampling

Convenience sampling attempts to obtain a sample of convenient elements. Often, respondents are selected because they happen to be in the right place at the right time.

use of students, and members of social organizations mall intercept interviews without qualifying the

respondents department stores using charge account lists “people on the street” interviews

Dr. Himani Gupta

Judgmental Sampling

Judgmental sampling is a form of convenience sampling in which the population elements are selected based on the judgment of the researcher.

test markets purchase engineers selected in industrial marketing

research bellwether precincts selected in voting behavior research expert witnesses used in court

Dr. Himani Gupta

Quota SamplingQuota sampling may be viewed as two-stage restricted judgmental sampling. The first stage consists of developing control categories, or quotas, of

population elements. In the second stage, sample elements are selected based on convenience

or judgment.

Population Samplecomposition composition

ControlCharacteristic Percentage Percentage NumberSexMale 48 48 480Female 52 52 520

____ ____ ____100 100 1000

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Snowball Sampling

In snowball sampling, an initial group of respondents is selected, usually at random.

After being interviewed, these respondents are asked to identify others who belong to the target population of interest.

Subsequent respondents are selected based on the referrals.

Dr. Himani Gupta

Types of Samples

In a probability sample, items in the sample are chosen on the basis of known probabilities.

Probability Samples

Simple Random Systematic Stratified Cluster

Dr. Himani Gupta

Simple Random Sampling…Example: A government income tax auditor must choose a sample of 40 of 1,000 returns to audit…

Extra #’s may be used if duplicate random numbers are generated

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Systematic Sampling

Convenient and relatively easy to administer

Population elements are an ordered sequence (at least, conceptually).

The first sample element is selected randomly from the first kpopulation elements.

Thereafter, sample elements are selected at a constant interval, k, from the ordered sequence frame.

k = N

n,

where:

n = sample size

N = population size

k = size of selection interval

Dr. Himani Gupta

Systematic Sampling: Example

Purchase orders for the previous fiscal year are serialized 1 to 10,000 (N = 10,000).

A sample of fifty (n = 50) purchases orders is needed for an audit.

k = 10,000/50 = 200 First sample element randomly selected from the

first 200 purchase orders. Assume the 45th purchase order was selected.

Subsequent sample elements: 245, 445, 645, . . .

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Stratified Random Sampling…A stratified random sample is obtained by separating the population into mutually exclusive sets, or strata, and then drawing simple random samples from each stratum.

Strata 1 : GenderMale

Female

Strata 2 : Age< 20

20-3031-4041-5051-60> 60

Strata 3 : Occupationprofessional

clericalblue collar

other

We can acquire about the total population, make inferences within a stratumor make comparisons across strata

Dr. Himani Gupta

Stratified Random Sampling…After the population has been stratified, we can use simple random sampling to generate the complete sample:

If we only have sufficient resources to sample 400 people total,we would draw 100 of them from the low income group…

…if we are sampling 1000 people, we’d draw50 of them from the high income group.

Dr. Himani Gupta

Divide population into two or more subgroups (called strata) according to some common characteristic.

A simple random sample is selected from each subgroup, with sample sizes 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.

Dr. Himani Gupta

Cluster Sampling Population is divided into several “clusters,” each

representative of the population.

A simple random sample of clusters is selected.

All items in the selected clusters can be used, or items can be chosen 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.

Dr. Himani Gupta

Cluster Sampling

•San Jose

•Boise

•Phoenix

• Denver

• Cedar Rapids

•Buffalo

•Louisville

•Atlanta

• Portland

• Milwaukee

• KansasCity

•SanDiego •Tucson

• Grand Forks• Fargo

•Sherman-Dension•Odessa-

Midland

•Cincinnati

• Pittsfield

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Types of Cluster SamplingCluster Sampling

One-StageSampling

MultistageSampling

Two-StageSampling

Simple ClusterSampling

ProbabilityProportionate

to Size Sampling

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Cluster Sampling Advantages

• More convenient for geographically dispersed populations• Reduced travel costs to contact sample elements• Simplified administration of the survey• Unavailability of sampling frame prohibits using other

random sampling methods Disadvantages

• Statistically less efficient when the cluster elements are similar

• Costs and problems of statistical analysis are greater than for simple random sampling

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Sample Size…

Numerical techniques for determining sample sizes will be described later, but suffice it to say that the larger the sample size is, the more accurate we can expect the sample estimates to be.

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Sampling and Non-Sampling Errors…

Two major types of error can arise when a sample of observations is taken from a population:sampling error and nonsampling error.

Sampling error refers to differences between the sample and the population that exist only because of the observations that happened to be selected for the sample.

Increasing the sample size will reduce this error.

Dr. Himani Gupta

Nonsampling Error…Nonsampling errors are more serious and are due to mistakes made in the acquisition of data or due to the sample observations being selected improperly. Three types of nonsampling errors:

Errors in data acquisitionNonresponse errorsSelection bias

Note: increasing the sample size will not reduce this type of error.

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Errors in data acquisition…

— incorrect measurements being taken because of faulty equipment,— mistakes made during transcription from primary sources,— inaccurate recording of data due to misinterpretation of terms, or— inaccurate responses to questions concerning sensitive issues.

…arises from the recording of incorrect responses, due to:

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Nonresponse Error… …refers to error (or bias) introduced when responses are not obtained from some members of the sample, i.e. the sample observations that are collected may not be representative of the target population.

As mentioned earlier, the Response Rate (i.e. the proportion of all people selected who complete the survey) is a key survey parameter and helps in the understanding in the validity of the survey and sources of nonresponse error.

Dr. Himani Gupta

Selection Bias…

…occurs when the sampling plan is such that some members of the target population cannot possibly be selected for inclusion in the sample.

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Sampling Distributions

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

For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students.

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Sampling DistributionsSample Mean Example

Suppose your population (simplified) was four people at your institution.

Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years)

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Sampling DistributionsSample Mean Example

Summary Measures for the Population Distribution:

214

24222018NX

μ i

2.236N

μ)(Xσ

2i

.3

.2

.1

018 20 22 24A B C D

P(x)

x

Uniform Distribution

Dr. Himani Gupta

Sampling DistributionsSample Mean Example

24,2424,2224,2024,1824

22,2422,2222,2022,1822

20,2420,2229,2020,1820

18,2418,2218,2018,1818

24222018

2nd Observation1st

Obs.

Now consider all possible samples of size n=2

2423222124

2322212022

2221201920

2120191818

24222018

2nd Observation1st

Obs.

16 Sample Means

16 possible samples (sampling with replacement)

Dr. Himani Gupta

Sampling DistributionsSample Mean Example

Sampling Distribution of All Sample Means

2423222124

2322212022

2221201920

2120191818

24222018

2nd Observation1st

Obs

16 Sample Means

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

(no longer uniform)

Sample Means Distribution

_

Dr. Himani Gupta

Sampling DistributionsSample Mean Example

2116

24211918NX

μ iX

1.5816

21)-(2421)-(1921)-(18

N)μX(

σ

222

2Xi

X

Summary Measures of this Sampling Distribution:

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Sampling DistributionsSample Mean ExamplePopulation

N = 4

1.58σ 21μ X X2.236σ 21μ

Sample Means Distributionn = 2

18 20 22 24A B C D

0

.1

.2

.3 P(X)

X 18 19 20 21 22 23 240

.1

.2

.3 P(X)

X_

_

Dr. Himani Gupta

Sampling DistributionsStandard Error

nσσX

Different samples of the same size from the same population will yield different sample means.

A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:

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

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Sampling DistributionsStandard Error: Normal Pop.

μμX nσσX

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

and

(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

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Sampling DistributionsZ Value: Normal Pop.

nσμ)X(

σ)μX(

ZX

X

Z-value for the sampling distribution of the sample mean:

where: = sample mean= population mean= population standard deviation

n = sample size

Xμσ

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Sampling DistributionsProperties: Normal Pop.

(i.e. is unbiased )

Normal Population Distribution

Normal Sampling Distribution (has the same mean)

μμx

xx

x

μ

xμ

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Sampling DistributionsProperties: Normal Pop.

For sampling with replacement: As n increases, decreasesxσ

Larger sample size

Smaller sample size

xμ

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Sampling DistributionsNon-Normal Population The Central Limit Theorem states that as the sample

size (that is, the number of values in each sample) gets large enough, the sampling distribution of the mean is approximately normally distributed. This is true regardless of the shape of the distribution of the individual values in the population.

Measures of the sampling distribution:

μμx nσσx

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Sampling DistributionsNon-Normal Population

Population Distribution

Sampling Distribution (becomes normal as n increases)

x

x

Larger sample size

Smaller sample size

xμ

μ

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Sampling DistributionsNon-Normal Population

For most distributions, n > 30 will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n > 15 will give a sampling distribution that is nearly normal

For normal population distributions, the sampling distribution of the mean is always normally distributed

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Sampling DistributionsExample

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

What is the probability that the sample mean is between 7.75 and 8.25?

Even if the population is not normally distributed, the central limit theorem can be used (n > 30).

So, the distribution of the sample mean is approximately normal with

8μx 0.5363

nσσx

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Sampling DistributionsExample

5.036

38-8.25

5.036

38-7.75

Z

Z

First, compute Z values for both 7.75 and 8.25.

0.38300.5)ZP(-0.5 8.25) μ P(7.75 X

Now, use the cumulative normal table to compute the correct probability.

Dr. Himani Gupta

Sampling DistributionsExample

= 2(.5000-.3085)

= 2(.1915)

= 0.3830

Z-0.5 0.5

Standardized Normal Distribution

0μz 7.75 8.25

Sampling Distribution

Sample

8μX x

Population Distribution

8μ X

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Sampling DistributionsThe Proportion

size sampleinterest of sticcharacteri thehaving sample in the ofnumber

nX itemsp

The proportion of the population having some characteristic is denoted π.

Sample proportion ( p ) provides an estimate of π:

0 ≤ p ≤ 1

p has a binomial distribution(assuming sampling with replacement from a finite population or without replacement from an infinite population)

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Sampling DistributionsThe Proportion Standard error for the proportion:

n)(1σp

n)(1σ

Zp

pp

Z value for the proportion:

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Sampling DistributionsThe Proportion: Example

If the true proportion of voters who support Proposition A is π = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45?

In other words, if π = .4 and n = 200, what is

P(.40 ≤ p ≤ .45) ?

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Sampling DistributionsThe Proportion: Example

.03464200

.4).4(1n

)(1σ

p

1.44)ZP(0.03464

.40.45Z.03464

.40.40P.45)P(.40

p

Find :

Convert to standardized normal:

pσ

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Sampling DistributionsThe Proportion: Example

Use cumulative normal table:

P(0 ≤ Z ≤ 1.44) = P(Z ≤ 1.44) – 0.5 = .4251

Z.45 1.44

.4251

Standardize

Sampling DistributionStandardized

Normal Distribution

.40 0p

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Problem 1

The mean expenditure per customer at a tire store is $85.00, with the standard deviation of $9.00. If a random sample of 40 customer is taken, what is the probability that the sample average expenditure per customer for this sample will be $87.00 or more?

Dr. Himani Gupta

Solution to Tire Store Example Population Parameters: Sample Size:

85 940

8787

87

,

( )

n

P X P Z

P Z

n

X

X

0793.9207.1

),41.1(141.1

409

8587

ZPZP

ZP

Dr. Himani Gupta

Graphic Solution to Tire Store Example

Z = X-n

87 85

940

21 42

1 41.

.

1

Z1.410

.5000

.4207

X

940

1 42.

X8785

.5000

.4207

Equal Areasof .0793

Dr. Himani Gupta

Problem 2

Suppose that during any hour in a large department store, the average number of the shoppers is 448, with a standard deviation of 21 shoppers. What is the probability that a random sample of 49 different shopping hours will yield a sample mean between 441 and 446 shoppers?

Dr. Himani Gupta

Graphic Solution Problem

Z= X-n

441 448

2149

2 33. Z= X-n

446 448

2149

0 67.

0

1

Z-2.33 -.67

.2486.4901

.2415

448

X 3

X441 446

.2486.4901

.2415

Dr. Himani Gupta

Sampling from a Finite Population without Replacement

In this case, the standard deviation of the distribution of sample means is smaller than when sampling from an infinite population (or from a finite population with replacement).

The correct value of this standard deviation is computed by applying a finite correction factor to the standard deviation for sampling from a infinite population.

If the sample size is less than 5% of the population size, the adjustment is unnecessary.

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Sampling from a Finite Population

Finite Correction Factor

Modified Z Formula

N nN 1

Z X

nN nN

1

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Finite Correction Factor for Selected Sample Sizes

Population Sample Sample % Value ofSize (N) Size (n) of Population Correction Factor

6,000 30 0.50% 0.9986,000 100 1.67% 0.9926,000 500 8.33% 0.9582,000 30 1.50% 0.9932,000 100 5.00% 0.9752,000 500 25.00% 0.866

500 30 6.00% 0.971500 50 10.00% 0.950500 100 20.00% 0.895200 30 15.00% 0.924200 50 25.00% 0.868200 75 37.50% 0.793

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Problem 3

A production company’s 350 hourly employees average 37.6 years of the age, with a standard deviation of 8.3 years. If a random sample of 45 hourly employees is taken, what is the probability that a sample will have an average of less than 40 years?

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Sampling Distribution of p Sample Proportion

Sampling Distribution• Approximately normal if nP > 5 and nQ > 5 (P is the

population proportion and Q = 1 - P.)• The mean of the distribution is P.• The standard deviation of the distribution is

:

p Xn

whereX

number of items in a sample that possess the characteristicn = number of items in the sample

P Qn

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Z Formula for Sample Proportions

p PZ

P Qn

wherepnPQ Pn Pn Q

: sample proportion

sample sizepopulation proportion

155

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Problem 4: If 10% of a population of parts is defective, what is the probability of randomly selecting 80 parts and finding that 12 or more parts are defective?

Population Parameters= .= -

Sample=

PQ P

nX

p Xn

P p P Z p

p

0 101 1 10 90

8012

1280

0 15

1515

. .

.

( . ).

P ZP Z

( . ). ( . ). ..

1 495 0 1 495 43190681

P

Z PP Q

n

. 15

P . .

(. )(. )15 1010 90

80

Z

P Z ..0 05

0 0335

Dr. Himani Gupta

Graphic Solution for Demonstration Problem 4

Z = . .

(. )(. ).

..p P

P Qn

0 15 0 10

10 9080

0 050 0335

1 49

1

Z1.490

.5000

.4319

.p 0 0335

p0.150.10

.5000

.4319

^