Slides by
John
Loucks
St. Edward’s
University
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
1
Chapter 7
Sampling and Sampling Distributions
◼ Selecting a Sample
◼ Point Estimation
◼ Introduction to Sampling Distributions
◼ Sampling Distribution of 𝑥ҧ
◼ Sampling Distribution of 𝑝ҧ
◼ Other Sampling Methods
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
2
Introduction
An element is the entity on which data are collected.
A population is a collection of all the elements of
interest.
A sample is a subset of the population.
The sampled population is the population from
which the sample is drawn.
A frame is a list of the elements that the sample will
be selected from.
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3
Introduction
The reason we select a sample is to collect data to
answer a research question about a population.
The sample results provide only estimates of the
values of the population characteristics.
The reason is simply that the sample contains only
a portion of the population.
With proper sampling methods, the sample results
can provide “good” estimates of the population
characteristics.
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4
Selecting a Sample
Sampling from a Finite Population
Sampling from an Infinite Population
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5
Sampling from a Finite Population
Finite populations are often defined by lists such as:
• Organization membership roster
• Credit card account numbers
• Inventory product numbers
A simple random sample of size n 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.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
6
Sampling from a Finite Population
◼ 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.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
7
Sampling from a Finite Population
Example: St. Andrew’s College
St. Andrew’s College received 900 applications for
admission in the upcoming year from prospective
students. The applicants were numbered, from 1 to
900, as their applications arrived. The Director of
Admissions would like to select a simple random
sample of 30 applicants.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
8
Sampling from a Finite Population
Example: St. Andrew’s College
Step 1: Assign a random number to each of the 900
applicants.
The random numbers generated by Excel’s
RAND function follow a uniform probability
distribution between 0 and 1.
Step 2: Select the 30 applicants corresponding to the
30 smallest random numbers.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
9
Sampling from a Finite Population Using Excel
Excel Formula Worksheet
A
Applicant
1
Number
2
1
3
2
4
3
5
4
6
5
7
6
8
7
9
8
B
Random
Number
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
Note: Rows 10-901 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
10
Sampling from a Finite Population Using Excel
Excel Value Worksheet
A
B
Applicant Random
1
Number Number
2
1
0.61021
3
2
0.83762
4
3
0.58935
5
4
0.19934
6
5
0.86658
7
6
0.60579
8
7
0.80960
9
8
0.33224
Note: Rows 10-901 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
11
Sampling from a Finite Population Using Excel
Put Random Numbers in Ascending Order
Step 1
Step 2
Step 3
Step 4
Select any cell in the range B2:B901
Click the Home tab on the Ribbon
In the Editing group, click Sort & Filter
Choose Sort Smallest to Largest
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
12
Sampling from a Finite Population Using Excel
Excel Value Worksheet (Sorted)
A
B
Applicant Random
1 Number Number
2
12
0.00027
3
773
0.00192
4
408
0.00303
5
58
0.00481
6
116
0.00538
7
185
0.00583
8
510
0.00649
9
394
0.00667
Note: Rows 10-901 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
13
Sampling from an Infinite Population
◼ Sometimes we want to select a sample, but find it is
not possible to obtain a list of all elements in the
population.
◼ As a result, we cannot construct a frame for the
population.
◼ Hence, we cannot use the random number selection
procedure.
◼ Most often this situation occurs in infinite population
cases.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
14
Sampling from an Infinite Population
Populations are often generated by an ongoing
process where there is no upper limit on the number of
units that can be generated.
◼ Some examples of on-going processes, with infinite
populations, are:
• parts being manufactured on a production line
• transactions occurring at a bank
• telephone calls arriving at a technical help desk
• customers entering a store
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
15
Sampling from an Infinite Population
◼ In the case of an infinite population, we must select
a random sample in order to make valid statistical
inferences about the population from which the
sample is taken.
A random sample from an infinite population is a
sample selected such that the following conditions
are satisfied.
• Each element selected comes from the population
of interest.
• Each element is selected independently.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
16
Point Estimation
Point estimation is a form of statistical inference.
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 𝑥ҧ as the point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
𝑝ҧ is the point estimator of the population proportion p.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
17
Point Estimation
Example: St. Andrew’s College
Recall that St. Andrew’s College received 900
applications from prospective students. The
application form contains a variety of information
including the individual’s Scholastic Aptitude Test
(SAT) score and whether or not the individual desires
on-campus housing.
At a meeting in a few hours, the Director of
Admissions would like to announce the average SAT
score and the proportion of applicants that want to
live on campus, for the population of 900 applicants.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
18
Point Estimation
Example: St. Andrew’s College
However, the necessary data on the applicants have
not yet been entered in the college’s computerized
database. So, the Director decides to estimate the
values of the population parameters of interest based
on sample statistics. The sample of 30 applicants is
selected using computer-generated random numbers.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
19
Point Estimation Using Excel
Excel Value Worksheet (Sorted)
A
B
C
Applicant Random
SAT
Score
1 Number Number
2
12
0.00027
1207
3
773
0.00192
1143
4
408
0.00303
1091
5
58
0.00481
1108
6
116
0.00538
1227
7
185
0.00583
982
8
510
0.00649
1363
9
1108
394
0.00667
D
On-Campus
Housing
No
Yes
Yes
No
Yes
Yes
Yes
No
Note: Rows 10-31 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
20
Point Estimation
𝑥ҧ as Point Estimator of
50,520
𝑥ҧ =
=
= 1684
30
30
σ 𝑥𝑖
s as Point Estimator of
𝑠=
σ(𝑥𝑖 − 𝑥)ҧ 2
=
29
210,512
= 85.2
29
𝑝ҧ as Point Estimator of p
𝑝ҧ = 20Τ30 = .67
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
21
Point Estimation
Once all the data for the 900 applicants were entered
in the college’s database, the values of the population
parameters of interest were calculated.
Population Mean SAT Score
σ 𝑥𝑖
𝜇=
= 1697
900
Population Standard Deviation for SAT Score
𝜎=
σ(𝑥𝑖 −𝜇)2
= 87.4
900
Population Proportion Wanting On-Campus Housing
𝑝 = 648/900 = .72
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22
Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
Point
Estimator
Point
Estimate
= Population mean
1697
𝑥ҧ = Sample mean
SAT score
1684
87.4
s = Sample standard deviation
for SAT score
85.2
.72
𝑝ҧ = Sample proportion wanting
campus housing
SAT score
= Population std.
deviation for
SAT score
p = Population proportion wanting
campus housing
.67
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23
Practical Advice
The target population is the population we want to
make inferences about.
The sampled population is the population from
which the sample is actually taken.
Whenever a sample is used to make inferences
about a population, we should make sure that the
targeted population and the sampled population
are in close agreement.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
24
Sampling Distribution of 𝑥ҧ
Process of Statistical Inference
Population
with mean
=?
The value of 𝑥ҧ is used to
make inferences about
the value of .
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean 𝑥ҧ .
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25
Sampling Distribution of 𝑥ҧ
The sampling distribution of 𝑥ҧ is the probability
distribution of all possible values of the sample
mean 𝑥.ҧ
• Expected Value of 𝑥ҧ
E(𝑥)ҧ =
where: = the population mean
When the expected value of the point estimator
equals the population parameter, we say the point
estimator is unbiased.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
26
Sampling Distribution of 𝑥ҧ
• Standard Deviation of 𝑥ҧ
We will use the following notation to define the
standard deviation of the sampling distribution of 𝑥.ҧ
𝜎𝑥ҧ = the standard deviation of 𝑥ҧ
= the standard deviation of the population
n = the sample size
N = the population size
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27
Sampling Distribution of 𝑥ҧ
• Standard Deviation of 𝑥ҧ
Finite Population
𝜎𝑥ҧ =
𝑁−𝑛
𝑁−1
𝜎
𝑛
Infinite Population
𝜎
𝜎𝑥ҧ =
𝑛
• A finite population is treated as being infinite if
n/N < .05.
•
(𝑁 − 𝑛)/(𝑁 − 1) is the finite population
correction factor.
• 𝜎𝑥ҧ is referred to as the standard error of the mean.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
28
Sampling Distribution of 𝑥ҧ
When the population has a normal distribution, the
sampling distribution of 𝑥ҧ is normally distributed
for any sample size.
In most applications, the sampling distribution of 𝑥ҧ
can be approximated by a normal distribution
whenever the sample is size 30 or more.
In cases where the population is highly skewed or
outliers are present, samples of size 50 may be
needed.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
29
Sampling Distribution of 𝑥ҧ
The sampling distribution of 𝑥ҧ can be used to
provide probability information about how close
the sample mean 𝑥ҧ is to the population mean .
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30
Central Limit Theorem
When the population from which we are selecting
a random sample does not have a normal distribution,
the central limit theorem is helpful in identifying the
shape of the sampling distribution of 𝑥.ҧ
CENTRAL LIMIT THEOREM
In selecting random samples of size n from a
population, the sampling distribution of the sample
mean 𝑥ҧ can be approximated by a normal
distribution as the sample size becomes large.
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31
Sampling Distribution of 𝑥ҧ
Example: St. Andrew’s College
Sampling
Distribution
of 𝑥ҧ for
SAT Scores
𝜎𝑥ҧ =
𝜎
𝑛
=
87.4
30
= 15.96
𝑥ҧ
𝐸 𝑥ҧ = 1697
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32
Sampling Distribution of 𝑥ҧ
Example: St. Andrew’s College
What is the probability that a simple random
sample of 30 applicants will provide an estimate of
the population mean SAT score that is within +/-10
of the actual population mean ?
In other words, what is the probability that 𝑥ҧ will
be between 1687 and 1707?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
33
Sampling Distribution of 𝑥ҧ
Example: St. Andrew’s College
Step 1: Calculate the z-value at the upper endpoint of
the interval.
z = (1707 - 1697)/15.96 = .63
Step 2: Find the area under the curve to the left of the
upper endpoint.
P(z < .63) = .7357
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in a license distributed with a certain product or service or otherwi ...

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