Chapter 7 Sampling Distributions

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© 2001 South-Western/Thomson Learning

Anderson u Sweeney u Williams

Anderson u Sweeney u Williams

u Slides Prepared by JOHN LOUCKS u

CONTEMPORARY

BUSINESSSTATISTICS

WITH MICROSOFT EXCEL

CONTEMPORARY

BUSINESSSTATISTICS

WITH MICROSOFT EXCEL

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Chapter 7Sampling and Sampling Distributions

Simple Random Sampling

Point Estimation Introduction to Sampling

Distributions Sampling Distribution of Sampling Distribution of Other Sampling Methods

xp

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Statistical Inference

The purpose of statistical inference is to obtain information about a population from information contained in a sample.

A population is the set of all the elements of interest.

A sample is a subset of the population. The sample results provide only estimates of

the values of the population characteristics. A parameter is a numerical characteristic of a

population. With proper sampling methods, the sample

results will provide “good” estimates of the population characteristics.

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Finite Population• A simple random sample from a finite

population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

• Replacing each sampled element before selecting subsequent elements is called sampling with replacement.

• Sampling without replacement is the procedure used most often.

• In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.

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Infinite Population• A simple random sample from an infinite

population is a sample selected such that the following conditions are satisfied.• Each element selected comes from the

same population.• Each element is selected independently.

• The population is usually considered infinite if it involves an ongoing process that makes listing or counting every element impossible.

• The random number selection procedure cannot be used for infinite populations.


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Point Estimation

In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter.

We refer to as the point estimator of the population mean .

s is the point estimator of the population standard deviation .

is the point estimator of the population proportion p.

x

p

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Sampling Error

The absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.

Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates.

The sampling errors are:for sample mean

|s - |s for sample standard deviation

for sample proportion

|| x

|| pp

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Example: St. Edward’s

St. Edward’s University receives 1,500 applications

annually from prospective students. The application

forms contain a variety of information including the

individual’s scholastic aptitude test (SAT) score and

whether or not the individual is an in-state resident.The director of admissions would like to know, at

least roughly, the following information:• the average SAT score for the applicants,

and• the proportion of applicants that are in-state

residents.We will now look at two alternatives for obtaining

the desired information.

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Alternative #1: Take a Census of 1,500 Applicants• SAT Scores

• Population Mean

• Population Standard Deviation

• In-State Applicants• Population Proportion

990500,1

ix

80500,1

)( 2

ix

72.500,1

080,1p


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Alternative #2: Take a Sample of 50 Applicants• Excel can be used to select a simple random

sample without replacement.• The process is based on random numbers

generated by Excel’s RAND function.• RAND function generates numbers in the

interval from 0 to 1.• Any number in the interval is equally likely.• The numbers are actually values of a

uniformly distributed random variable.


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Using Excel to Select a Simple Random Sample• 1500 random numbers are generated, one

for each applicant in the population.• Then we choose the 50 applicants

corresponding to the 50 smallest random numbers as our sample.

• Each of the 1500 applicants have the same probability of being included.

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Using Excel to Selecta Simple Random Sample

Formula Worksheet

A B C

1 SAT Score In-StateRandom Number

2 1008 Yes =RAND()3 1025 No =RAND()4 952 Yes =RAND()5 1090 Yes =RAND()6 1127 Yes =RAND()7 1015 No =RAND()8 965 Yes =RAND()9 1161 No =RAND()

Note: Rows 10-1501 are not shown.

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Value Worksheet

A B C


2 1008 Yes 0.381843 1025 No 0.080374 952 Yes 0.255155 1090 Yes 0.822256 1127 Yes 0.387007 1015 No 0.529998 965 Yes 0.279629 1161 No 0.28245


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Put Random Numbers in Ascending Order• Step 1 Select cells A2:A1501• Step 2 Select the Data pull-down menu• Step 3 Choose the Sort option• Step 4 When the Sort dialog box appears:

Choose Random Numbersin the Sort by text box

Choose Ascending Click OK

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Value Worksheet (Sorted)

A B C


2 1107 No 0.000273 1043 Yes 0.001924 991 Yes 0.003035 1008 No 0.004816 1127 Yes 0.005387 982 Yes 0.005838 1163 Yes 0.006499 1008 No 0.00667


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Point Estimates• as Point Estimator of

• s as Point Estimator of

• as Point Estimator of p

Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

x

p

xxi 50

49 85050

997,

sx xi ( ) ,

.2

49277 09749

75 2

p 34 50 68.


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Sampling Distribution of

The sampling distribution of is the probability distribution of all possible values of the sample mean .

Expected Value of E(x) =

where: = the population mean

x

x

x

x

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Standard Deviation of

Finite Population Infinite Population

• A finite population is treated as being infinite if n/N < .05.

• is the finite correction factor.• is referred to as the standard error of the

mean.

x

x n

N nN

( )1

x n

( ) / ( )N n N 1

x

Sampling Distribution of x

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If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal probability distribution.

When the simple random sample is small (n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal probability distribution.

x

x

Sampling Distribution of x

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Sampling Distribution of for the SAT Scoresx


E x( ) 990

x n 80

5011 3.

x

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Sampling Distribution of for the SAT ScoresWhat is the probability that a simple

random sample of 50 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean ?

In other words, what is the probability that will be between 980 and 1000?

x


x

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Sampling Distribution of for the SAT Scores

Using the standard normal probability table with z = 10/11.3 = .88, we have area = (.3106)(2) = .6212.

x

Sampling distribution of

Sampling distribution of x

1000980 990

Area = .3106Area = .3106


x

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The sampling distribution of is the probability distribution of all possible values of the sample proportion .

Expected Value of

where:p = the population proportion

Sampling Distribution of p

p

p

p

E p p( )

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Sampling Distribution of

Standard Deviation of

Finite Population Infinite Population

• is referred to as the standard error of the proportion.

p

p

pp pn

N nN

( )11

pp pn

( )1

p

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Sampling Distribution of for In-State Residents

The normal probability distribution is an acceptable approximation since np = 50(.72) = 36 > 5 andn(1 - p) = 50(.28) = 14 > 5.

p

p

. ( . )

.72 1 72

500635

E p p( ) . 72


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What is the probability that a simple random sample of 50 applicants will provide an estimate of the population proportion of in-state residents that is within plus or minus .05 of the actual population proportion?

In other words, what is the probability that will be between .67 and .77?

p


p

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For z = .05/.0635 = .79, the area = (.2852)(2) = .5704.

The probability is .5704 that the sample proportion will

be within +/-.05 of the actual population proportion.



0.770.67 0.72

Area = .2852Area = .2852

p

p


p

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Other Sampling Methods

Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling

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Stratified Random Sampling

The population is first divided into groups of elements called strata.

Each element in the population belongs to one and only one stratum.

Best results are obtained when the elements within each stratum are as much alike as possible (i.e. homogeneous group).

A simple random sample is taken from each stratum.

Formulas are available for combining the stratum sample results into one population parameter estimate.

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Stratified Random Sampling

Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size.

Example: The basis for forming the strata might be department, location, age, industry type, etc.

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Cluster Sampling

The population is first divided into separate groups of elements called clusters.

Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).

A simple random sample of the clusters is then taken.

All elements within each sampled (chosen) cluster form the sample.

… continued

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Cluster Sampling

Advantage: The close proximity of elements can be cost effective (I.e. many sample observations can be obtained in a short time).

Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling.

Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.

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Systematic Sampling

If a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population.

We randomly select one of the first n/N elements from the population list.

We then select every n/Nth element that follows in the population list.

This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.

… continued

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Systematic Sampling

Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used.

Example: Selecting every 100th listing in a telephone book after the first randomly selected listing.

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Convenience Sampling

It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.

The sample is identified primarily by convenience.

Advantage: Sample selection and data collection are relatively easy.

Disadvantage: It is impossible to determine how representative of the population the sample is.

Example: A professor conducting research might use student volunteers to constitute a sample.

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Judgment Sampling

The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population.

It is a nonprobability sampling technique. Advantage: It is a relatively easy way of

selecting a sample. Disadvantage: The quality of the sample

results depends on the judgment of the person selecting the sample.

Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

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End of Chapter 7

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Chapter 7 Sampling Distributions