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Chapter 7
Student Lecture Notes
7-1
Department of Quantitative Methods & Information Systems
Business Statistics:
Chapter 7
Introduction to
Sampling Distributions
QMIS 220
Dr. Mohammad Zainal
Chapter Goals
After completing this chapter, you should be
able to:





Define the concept of sampling error
Determine the mean and standard deviation
_ for the
sampling distribution of the sample mean, x
Determine the mean and standard deviation for the
_
sampling distribution of the sample proportion, p
Describe the Central Limit Theorem and its importance
_
_
Apply sampling distributions for both x and p
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-2
Chapter 7
Student Lecture Notes
7-2
Review: Inferential Statistics

Inferential statistics



Drawing conclusions and/or making decisions
concerning a population based only on
sample data
Consists of methods that use sample results
to help make decisions or predictions about a
population.
Elections
QMIS 220, by Dr. M. Zainal
Chap 7-3
Review: Inferential Statistics
Sample statistics
(known)
Inference
Sample
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Population parameters
(unknown, but can
be estimated from
sample evidence)
Population
Chap 7-4
Chapter 7
Student Lecture Notes
7-3
Review: Inferential Statistics
Drawing conclusions and/or making decisions
concerning a population based on sample results.

Estimation


e.g., Estimate the population mean
weight using the sample mean
weight
Hypothesis Testing

e.g., Use sample evidence to test
the claim that the population mean
weight is 120 pounds
QMIS 220, by Dr. M. Zainal
Chap 7-5
Review: Key Definitions

A population is the entire collection of things
under consideration


A parameter is a summary measure computed to
describe a characteristic of the population
A sample is a portion of the population
selected for analysis

A statistic is a summary measure computed to
describe a characteristic of the sample
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-6
Chapter 7
Student Lecture Notes
7-4
Review: Population vs. Sample
Population
a b
Sample
cd
b
ef gh i jk l m n
o p q rs t u v w
x y
z
c
gi
o
n
r
u
y
QMIS 220, by Dr. M. Zainal
Chap 7-7
Review: Why Sample?

Less time consuming than a census

Less costly to administer than a census

It is possible to obtain statistical results of a
sufficiently high precision based on samples.
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-8
Chapter 7
Student Lecture Notes
7-5
Review: Sampling Techniques
Sampling Techniques
Nonstatistical Sampling
Statistical Sampling
Simple
Random
Convenience
Systematic
Judgment
Cluster
Stratified
QMIS 220, by Dr. M. Zainal
Chap 7-9
Review: Statistical Sampling

Items of the sample are chosen based on
known or calculable probabilities
Statistical Sampling
(Probability Sampling)
Simple Random
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Stratified
Systematic
Cluster
Chap 7-10
Chapter 7
Student Lecture Notes
7-6
Simple Random Sampling



Every possible sample of a given size has an
equal chance of being selected
Selection may be with replacement or without
replacement
The sample can be obtained using a table of
random numbers or computer random number
generator
QMIS 220, by Dr. M. Zainal
Chap 7-11
Stratified Random Sampling



Divide population into subgroups (called strata)
according to some common characteristic
Select a simple random sample from each
subgroup
Combine samples from subgroups into one
Population
Divided
into 4
strata
Sample
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-12
Chapter 7
Student Lecture Notes
7-7
Systematic Random Sampling




Decide on sample size: n
Divide frame of N individuals into groups of k
individuals: k=N/n
Randomly select one individual from the 1st
group
Select every kth individual thereafter
N = 64
n=8
First Group
k=8
QMIS 220, by Dr. M. Zainal
Chap 7-13
Cluster Sampling


Divide population into several “clusters,” each
representative of the population
Select a simple random sample of clusters

All items in the selected clusters can be used, or items can be
chosen from a cluster using another probability sampling
technique
Population
divided into
16 clusters.
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Randomly selected
clusters for sample
Chap 7-14
Chapter 7
Student Lecture Notes
7-8
Examples of poor samplings
The technique of sampling has been widely
used, both properly and improperly, in the area of
politics.

During the 1936 presidential race where the
Literary Digest predicted Alf Landon to win the
election over Franklin D. Roosevelt.

QMIS 220, by Dr. M. Zainal
Chap 7-15
Sampling Error
So far, we have stressed the benefits of drawing
a sample from a population.

However, in statistics, as in life, there's no such
thing as a free lunch.

By sampling, we expose ourselves to errors that
can lead to inaccurate conclusions about the
population.

The type of error that a statistician is most
concerned about is called sampling error.

QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-16
Chapter 7
Student Lecture Notes
7-9
Sampling Error

Sample Statistics are used to estimate
Population Parameters
ex: X is an estimate of the population mean, μ

Problems:


Different samples provide different estimates of
the population parameter
Sample results have potential variability, thus
sampling error exits
QMIS 220, by Dr. M. Zainal
Chap 7-17
Sampling Error
As the entire population is rarely measured, the
sampling error cannot be directly calculated.

With inferential statistics, we'll be able to assign
probabilities to certain amounts of sampling error
later.

It occurs when we select a sample that is not a
perfect match to the entire population.

Sampling errors are a small price to pay to avoid
measuring an entire population.

QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-18
Chapter 7
Student Lecture Notes
7-10
Sampling Error
One way to reduce the sampling error of a
statistical study is to increase the size of the
sample.




In general, the larger the sample size, the
smaller the sampling error.
If you increase the sample size until it reaches
the size of the population, then the sampling
error will be reduced to 0.
But in doing so, we lose the benefits of sampling.
QMIS 220, by Dr. M. Zainal
Chap 7-19
Calculating Sampling Error

Sampling Error:
The difference between a value (a statistic)
computed from a sample and the corresponding
value (a parameter) computed from a population
Example: (for the mean)
SamplingError  x - μ
where:
x  sample mean
μ  population mean
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-20
Chapter 7
Student Lecture Notes
7-11
Review
Population mean:

μ
x
N
i
Sample Mean:
x
x
i
n
where:
μ = Population mean
x = sample mean
xi = Values in the population or sample
N = Population size
n = sample size
QMIS 220, by Dr. M. Zainal
Chap 7-21
Example
If the population mean is μ = 98.6 degrees
and a sample of n = 5 temperatures yields a
sample mean of x = 99.2 degrees, then the
sampling error is
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-22
Chapter 7
Student Lecture Notes
7-12
Sampling Errors



Different samples will yield different sampling
errors
The sampling error may be positive or negative
( x may be greater than or less than μ)
The expected sampling error decreases as the
sample size increases
QMIS 220, by Dr. M. Zainal
Chap 7-23
Sampling Distribution

A sampling distribution is a
distribution of the possible values of
a statistic for a given size sample
selected from a population
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-24
Chapter 7
Student Lecture Notes
7-13
Developing a
Sampling Distribution

Assume there is a population …

Population size N=4


Random variable, x,
is age of individuals
A
B
C
D
Values of x: 18, 20,
22, 24 (years)
QMIS 220, by Dr. M. Zainal
Chap 7-25
Developing a
Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-26
Chapter 7
Student Lecture Notes
7-14
Developing a
Sampling Distribution
(continued)
Now consider all possible samples of size n=2
QMIS 220, by Dr. M. Zainal
Chap 7-27
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-28
Chapter 7
Student Lecture Notes
7-15
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
QMIS 220, by Dr. M. Zainal
Chap 7-29
Comparing the Population with
its Sampling Distribution
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-30
Chapter 7
Student Lecture Notes
7-16
Properties of a Sampling
Distribution
For any population,


the average value of all possible sample means computed from
all possible random samples of a given size from the population
is equal to the population mean:
μx  μ

Theorem 1
The standard deviation of the possible sample means
computed from all random samples of size n is equal to the
population standard deviation divided by the square root of the
sample size:
σx 
σ
n
Theorem 2
QMIS 220, by Dr. M. Zainal
Chap 7-31
If the Population is Normal
If a population is normal with mean μ and
standard deviation σ, the sampling distribution
of
x
is also normally distributed with
μx  μ
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
and
σx 
σ
n
Theorem 3
Chap 7-32
Chapter 7
Student Lecture Notes
7-17
z-value for Sampling Distribution
of x

Z-value for the sampling distribution of
z
where:
x:
(x  μ)
σ
n
x = sample mean
μ = population mean
σ = population standard deviation
n = sample size
QMIS 220, by Dr. M. Zainal
Chap 7-33
Finite Population Correction

Apply the Finite Population Correction if:
 the sample is large relative to the population
(n is greater than 5% of N)
and…
 Sampling is without replacement
Then
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
z
(x  μ)
σ Nn
n N 1
Chap 7-34
Chapter 7
Student Lecture Notes
7-18
Sampling Distribution Properties

The sample mean is an unbiased estimator
Normal Population
Distribution
x
μ
μx  μ
Normal Sampling
Distribution
(has the same mean)
μx
x
QMIS 220, by Dr. M. Zainal
Chap 7-35
Sampling Distribution Properties
(continued)

The sample mean is a consistent estimator
(the value of x becomes closer to μ as n increases):
Population
x
Small
sample size
As n increases,
x
σ x  σ/ n decreases
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Larger
sample size
μ
x
Chap 7-36
Chapter 7
Student Lecture Notes
7-19
If the Population is not Normal

We 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
…and the sampling distribution will have
μx  μ
and
σx 
σ
n
Theorem 4
QMIS 220, by Dr. M. Zainal
Chap 7-37
Central Limit Theorem
As the
sample
size gets
large
enough…
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
n↑
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
x
Chap 7-38
Chapter 7
Student Lecture Notes
7-20
If the Population is not Normal
(continued)
Population Distribution
Sampling distribution
properties:
Central Tendency
μx  μ
Variation
σx 
σ
n
x
μ
Sampling Distribution
(becomes normal as n increases)
Larger
sample
size
Smaller
sample size
(Sampling with replacement)
QMIS 220, by Dr. M. Zainal
μx
x
Chap 7-39
How Large is Large Enough?



For most distributions, n > 30 will give a
sampling distribution that is nearly normal
For fairly symmetric distributions, n > 15 is
sufficient
For normal population distributions, the
sampling distribution of the mean is always
normally distributed
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-40
Chapter 7
Student Lecture Notes
7-21
Example


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 7.8 and 8.2?
QMIS 220, by Dr. M. Zainal
Chap 7-41
Example
(continued)
Solution:
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-42
Chapter 7
Student Lecture Notes
7-22
Example
(continued)
Solution (continued) -- find z-scores:
QMIS 220, by Dr. M. Zainal
Chap 7-43
Population Proportions, π
π = the proportion of the population having
some characteristic

Sample proportion ( p ) provides an estimate
of π :
p

If two outcomes, p has a binomial distribution
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
x
number of successesin the sample

n
samplesize
Chap 7-44
Chapter 7
Student Lecture Notes
7-23
Sampling Distribution of p

Approximated by a
normal distribution if:

Sampling Distribution
P( p )
.3
.2
.1
0
nπ  5
n(1 π)  5
0
.2
where
μp  π
and
σp 
.4
.6
8
1
p
π(1 π)
n
(where π = population proportion)
QMIS 220, by Dr. M. Zainal
Chap 7-45
z-Value for Proportions
Standardize p to a z value with the formula:
z

pπ

σp
pπ
π(1  π)
n
If sampling is without replacement
and n is greater than 5% of the
population size, then σ p must use
the finite population correction
factor:
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
σp 
π(1  π) N  n
n
N 1
Chap 7-46
Chapter 7
Student Lecture Notes
7-24
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?
i.e.: if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
QMIS 220, by Dr. M. Zainal
Chap 7-47
Example
(continued)

QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?
Chap 7-48
Chapter 7
Student Lecture Notes
7-25
Example
(continued)
if π = .4 and n = 200, what is
P(.40 ≤ p ≤ .45) ?

QMIS 220, by Dr. M. Zainal
Chap 7-49
Chapter Summary



Discussed sampling error
Introduced sampling distributions
Described the sampling distribution of the mean





For normal populations
Using the Central Limit Theorem
Described the sampling distribution of a
proportion
Calculated probabilities using sampling
distributions
Discussed sampling from finite populations
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-50
Chapter 7
Student Lecture Notes
7-26
Copyright
The materials of this presentation were mostly
taken from the PowerPoint files accompanied
Business Statistics: A Decision-Making Approach,
7e © 2008 Prentice-Hall, Inc.
QMIS 220, by Dr. M. Zainal
QMIS 220, by Dr. M. Zainal
Chap 7-51