Sampling distribution

Business Statistics: Communicating with Numbers
By Sanjiv Jaggia and Alison Kelly
McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

7-2
Chapter 7 Learning Objectives (LOs)
LO 7.1: Differentiate between a population
parameter and a sample statistic.
LO 7.5: Describe the properties of the sampling
distribution of the sample mean.

7-3
Chapter 7 Learning Objectives (LOs)
LO 7.6: Explain the importance of the central
limit theorem.
LO 7.7: Describe the properties of the sampling
distribution of the sample proportion.
LO 7.8: Use a finite population correction
factor.

7-4
7.1 Sampling
LO 7.1 Differentiate between a population parameter and sample statistic.
 Population—consists of all items of interest
in a statistical problem.
 Population Parameter is unknown.
 Sample—a subset of the population.
 Sample Statistic is calculated from sample and
used to make inferences about the population.
 Bias—the tendency of a sample statistic to
systematically over- or underestimate a
population parameter.

7-5
7.1 Sampling
 Classic Case of a “Bad” Sample: The Literary
Digest Debacle of 1936
 During the1936 presidential election, the Literary Digest
predicted a landslide victory for Alf Landon over Franklin
D. Roosevelt (FDR) with only a 1% margin of error.
 They were wrong! FDR won in a landslide election.
 The Literary Digest had committed selection bias by
randomly sampling from their own subscriber/
membership lists, etc.
 In addition, with only a 24% response rate, the Literary
Digest had a great deal of non-response bias.
LO 7.2 Explain common sample biases.

7-6
7.1 SamplingLO 7.2
 Selection bias—a systematic exclusion of certain
groups from consideration for the sample.
 The Literary Digest committed selection bias by excluding
a large portion of the population (e.g., lower income
voters).
 Nonresponse bias—a systematic difference in
preferences between respondents and non-
respondents to a survey or a poll.
 The Literary Digest had only a 24% response rate. This
indicates that only those who cared a great deal about the
election took the time to respond to the survey. These
respondents may be atypical of the population as a whole.

7-7
7.1 Sampling
 Sampling Methods
 Simple random sample is a sample of n
observations which has the same probability of
being selected from the population as any other
sample of n observations.
 Most statistical methods presume simple
random samples.
 However, in some situations other sampling
methods have an advantage over simple
random samples.
LO 7.3 Describe simple random sampling.

7-8
7.1 SamplingLO 7.3
 Example: In 1961, students invested 24 hours per
week in their academic pursuits, whereas today’s
students study an average of 14 hours per week.
 A dean at a large university in California wonders if this
trend is reflective of the students at her university. The
university has 20,000 students and the dean would like a
sample of 100. Use Excel to draw a simple random
sample of 100 students.
 In Excel, choose
Formulas > Insert function >
RANDBETWEEN and input
the values shown here.

7-9
Simple Random Sample: Using Table of
Random Numbers
A population consists of 845 employees of Nitra Industries. A
sample of 52 employees is to be selected from that
population.
A more convenient method of selecting a random sample is
to use the identification number of each employee and a
table of random numbers such as the one in Appendix B.6.

7-10
Systematic Random Sampling
EXAMPLE
A population consists of 845 employees of Nitra Industries. A sample
of 52 employees is to be selected from that population.
First, k is calculated as the population size divided by the sample
size. For Nitra Industries, we would select every 16th (845/52)
employee list. If k is not a whole number, then round down.
Random sampling is used in the selection of the first name. Then,
select every 16th name on the list thereafter.
Systematic Random Sampling: The items or individuals of the
population are arranged in some order. A random starting point is
selected and then every kth member of the population is selected for
the sample.

7-11
7.1 Sampling
 Stratified Random Sampling
 Divide the population into mutually exclusive and
collectively exhaustive groups, called strata.
 Randomly select observations from each stratum,
which are proportional to the stratum’s size.
 Advantages:
 Guarantees that the each population subdivision is
represented in the sample.
 Parameter estimates have greater precision than those
estimated from simple random sampling.
LO 7.4 Distinguish between stratified random sampling and cluster sampling.

7-12
Stratified Random Sampling
Stratified Random Sampling: A population is first divided into
subgroups, called strata, and a sample is selected from each stratum.
Useful when a population can be clearly divided in groups based on
some characteristics
Suppose we want to study the advertising
expenditures for the 352 largest companies
in the United States to determine whether
firms with high returns on equity (a measure
of profitability) spent more of each sales
dollar on advertising than firms with a low
return or deficit.
To make sure that the sample is a fair
representation of the 352 companies, the
companies are grouped on percent return
on equity and a sample proportional to the
relative size of the group is randomly
selected.

7-13
7.1 SamplingLO 7.4
 Cluster Sampling
 Divide population into mutually exclusive and
collectively exhaustive groups, called clusters.
 Randomly select clusters.
 Sample every observation in those randomly
selected clusters.
 Advantages and disadvantages:
 Less expensive than other sampling methods.
 Less precision than simple random sampling or
stratified sampling.
 Useful when clusters occur naturally in the population.

7-14
Cluster Sampling
Cluster Sampling: A population is divided into clusters using naturally
occurring geographic or other boundaries. Then, clusters are
randomly selected and a sample is collected by randomly selecting
from each cluster.
Suppose you want to determine the views
of residents in Oregon about state and
federal environmental protection policies.
Cluster sampling can be used by
subdividing the state into small units—
either counties or regions, select at
random say 4 regions, then take samples
of the residents in each of these regions
and interview them. (Note that this is a
combination of cluster sampling and
simple random sampling.)

7-15
7.1 SamplingLO 7.4
 Stratified versus Cluster Sampling
 Stratified Sampling
 Sample consists of
elements from
each group.
 Preferred when the
objective is to
increase precision.
 Cluster Sampling
 Sample consists of
elements from the
selected groups.
 Preferred when
the objective is to
reduce costs.

7-16
7.2 The Sampling Distribution of the Means
 Population is described by parameters.
 A parameter is a constant, whose value may be unknown.
 Only one population.
 Sample is described by statistics.
 A statistic is a random variable whose value depends on
the chosen random sample.
 Statistics are used to make inferences about the
population parameters.
 Can draw multiple random samples of size n.
LO 7.5 Describe the properties of the sampling distribution of the
sample mean.

7-17
Methods of Probability Sampling
 In nonprobability sample inclusion in the
sample is based on the judgment of the
person selecting the sample.
 The sampling error is the difference
between a sample statistic and its
corresponding population parameter.

7-18
7.2 The Sampling Distribution of the
Sample Mean
 Estimator
 A statistic that is used to estimate a population
parameter.
 For example, , the mean of the sample, is an
estimator of m, the mean of the population.
 Estimate
 A particular value of the estimator.
 For example, the mean of the sample is an
estimate of m, the mean of the population.
LO 7.5
X
x

7-19
Sample Mean
 Sampling Distribution of the Mean
 Each random sample of size n drawn from the
population provides an estimate of m—the sample
mean .
 The sampling distribution of the sample mean is a
probability distribution consisting of all possible
sample means of a given sample size selected
from a population.
LO 7.5
X
x

7-20
Sample Mean
 Example
LO 7.5
Random Variable
X1 X2 X3 X4
Mean of
X
6 10 8 4 5.57
5 10 4 3 5.71
1 8 4 3 6.36
4 1 6 2 4.07
6 6 8 4
7 7 8 6
1 5 10 5
5 5 9 1
4 6 4 2
7 4 9 5
8 5 8 6
9 2 7 7
9 1 2 3
6 10 2 6
Means 5.57 5.71 6.36 4.07 5.43
One simple random sample
drawn from the population—a
single distribution of values of X.
Means from each
distribution (random
draw) from the
population.
A distribution of
means from each
random draw from
the population.

7-21
Sample Mean
 Example: Draw a sampling distribution of
the sample mean from a population of
X={1,2,3,4,5,6} from a sample of n=3
without replacement.
 Process:
 List all possible samples
 Calculate each mean of all possible samples
 Construct the distribution of the sample means
LO 7.5

7-22
Sample Mean
 The Expected Value and Standard Deviation
of the Sample Mean
 Expected Value
 The expected value of X,
 The expected value of the sample mean,
LO 7.5
 E X m
   E X E X m 

7-23
Sample Mean
 The Expected Value and Standard Deviation
of the Sample Mean
 Variance of X
 Standard Deviation
 of X
 of
LO 7.5
  2
Var X 
X
Where n is the sample size.
Also known as the Standard
Error of the Mean.
  2
SD X   
 SD X
n



7-24
Sample Mean
 The Central Limit Theorem
 For any population X with expected value m and standard
deviation , the sampling distribution of X will be
approximately normal if the sample size n is sufficiently
large.
 As a general guideline, the normal distribution
approximation is justified when n > 30.
 As before, if is approximately
normal, then we can transform it to
LO 7.6 Explain the importance of the central limit theorem.
X X
Z
n
m




7-25
Central Limit Theorem
 If the population follows a normal probability distribution, then for
any sample size the sampling distribution of the sample mean will
also be normal.
 If the population distribution is symmetrical (but not normal), the
normal shape of the distribution of the sample mean emerge with
samples as small as 10.
 If a distribution that is skewed or has thick tails, it may require
samples of 30 or more to observe the normality feature.
 The mean of the sampling distribution equal to μ and the variance
equal to σ2/n.
CENTRAL LIMIT THEOREM If all samples of a particular size are
selected from any population, the sampling distribution of the sample
mean is approximately a normal distribution. This approximation
improves with larger samples.

7-27
Using the Sampling
Distribution of the Sample Mean (Sigma Known)
 If a population follows the normal distribution, the
sampling distribution of the sample mean will also
follow the normal distribution.
 If the shape is known to be nonnormal, but the sample
contains at least 30 observations, the central limit
theorem guarantees the sampling distribution of the
mean follows a normal distribution.
 To determine the probability a sample mean falls within
a particular region, use:
n
X
z

m


7-28
 If the population does not follow the normal
distribution, but the sample is of at least 30
observations, the sample means will follow
the normal distribution.
 To determine the probability a sample mean
falls within a particular region, use:
ns
X
t
m

Using the Sampling
Distribution of the Sample Mean (Sigma Unknown)

7-29
Sample Mean
 Example: Given that m = 16 inches and  = 0.8
inches, determine the following:
 What is the expected value and the standard
deviation of the sample mean derived from a
random sample of
 2 pizzas
 4 pizzas
LO 7.5
  16E X m    0.8
0.57
2
SD X
n

  
  16E X m    0.8
0.40
4
SD X
n

  

7-30
Sample Mean
 Sampling from a Normal Distribution
 For any sample size n, the sampling distribution
of is normal if the population X from which the
sample is drawn is normally distributed.
 If X is normal, then we can transform it into the
standard normal random variable as:
LO 7.5
X
 
 
X E X X
Z
SD X n
m

 
 
For a sampling
distribution.
 
 
x E X x
Z
SD X
m

 
 
For a distribution of
the values of X.

7-31
Sample Mean
LO 7.5
 Example: Given that m = 16 inches and  = 0.8
inches, determine the following:
 What is the probability that a randomly selected pizza is
less than 15.5 inches?

 What is the probability that 2 randomly selected pizzas
average less than 15.5 inches?

15.5 16
0.63
0.8
x
Z
m

 
   
( 15.5) ( 0.63)
0.2643 or 26.43%
P X P Z   

( 15.5) ( 0.88)
0.1894 or 18.94%
P X P Z   

15.5 16
0.88
0.8 2
x
Z
n
m

 
   

7-32
 Example: From the introductory case, Anne wants
to determine if the marketing campaign has had a
lingering effect on the amount of money customers
spend on iced coffee.
 Before the campaign, m = $4.18 and  = $0.84. Based on 50
customers sampled after the campaign, m = $4.26.
 Let’s find . Since n > 30, the central limit
theorem states that is approximately normal. So,
 4.26P X 
X
Sample Mean
LO 7.6
 
 
4.26 4.18
4.26
0.84 50
0.67 1 0.7486 0.2514
X
P X P Z P Z
n
P Z
m

    
       
   
    

7-33
Sample Proportion
LO 7.7 Describe the properties of the sampling distribution of the sample
proportion.
 Estimator
 Sample proportion is used to estimate the
population parameter p.
 Estimate
 A particular value of the estimator .
P
p

7-34
Sample Proportion
LO 7.7
 The Expected Value and Standard
Deviation of the Sample Proportion
 Expected Value
 The expected value of ,
 The standard deviation of ,
 E P p
P
P
   1p p
SD P
n



7-35
Sample Proportion
LO 7.7
 The Central Limit Theorem for the Sample
Proportion
 For any population proportion p, the sampling
distribution of is approximately normal if the
sample size n is sufficiently large .
 As a general guideline, the normal distribution
approximation is justified when
np > 5 and n(1  p) > 5.
P

7-36
Sample Proportion
LO 7.7
Proportion
 If is normal, we can transform it into the
standard normal random variable as
 Therefore any value on
has a corresponding value
z on Z given by
P
 
   1
P E P P p
Z
SD P p p
n
 
 

Pp
 


1
p p
Z
p p
n

7-37
Proportion
Sample Proportion
LO 7.7
Sampling distribution of
when the population proportion
is p = 0.10.
P Sampling distribution of
when the population proportion
is p = 0.30.
P

7-38
 Example: From the introductory case, Anne wants
to determine if the marketing campaign has had a
lingering effect on the proportion of customers who
are women and teenage girls.
 Before the campaign, p = 0.43 for women and p = 0.21 for
teenage girls. Based on 50 customers sampled after the
campaign, p = 0.46 and p = 0.34, respectively.
 Let’s find . Since n > 30, the central limit
theorem states that is approximately normal.
Sample Proportion
LO 7.7
 0.46P P 
P

7-39
Sample Proportion
LO 7.7
 
   
 
0.46 0.43
0.46
1 0.43 1 0.43
50
0.43 1 0.6664 0.3336
p p
P P P Z P Z
p p
n
P Z
  
  
       
   
  
   
    

Sampling distribution

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