Slides by
JOHN
LOUCKS
St. Edward’s
University
© 2009 Thomson South-Western. All Rights Reserved
Slide 1
Chapter 7, Part A
Sampling and Sampling Distributions
 Simple Random Sampling
 Point Estimation
 Introduction to Sampling Distributions
 Sampling Distribution of
x
© 2009 Thomson South-Western. All Rights Reserved
Slide 2
Statistical Inference
The purpose of statistical inference is to obtain
information about a population from information
contained in a sample.
An element is the entity on which data are collected.
A population is the set of all the elements of interest.
A sample is a subset of the population.
© 2009 Thomson South-Western. All Rights Reserved
Slide 3
Statistical Inference
The sample results provide only estimates of the
values of the population characteristics.
With proper sampling methods, the sample results
can provide “good” estimates of the population
characteristics.
A parameter is a numerical characteristic of a
population.
© 2009 Thomson South-Western. All Rights Reserved
Slide 4
Statistical Inference
A 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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 5
Statistical Inference
When making inferences about a population based
on a sample, it is important to have a close
correspondence between the sampled population
and the target population.
The target population is the population we want to
make inferences about.
© 2009 Thomson South-Western. All Rights Reserved
Slide 6
Simple Random 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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 7
Simple Random 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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 8
Simple Random Sampling
From an Infinite Population


Sometimes the sampled population is such that a
frame cannot be constructed.
One such situation is sampling from an ongoing
process where the conceptual population is infinite.

Another situation is sampling from a possibly very
large population where it is not possible, or perhaps
feasible, to identify all the elements in the population.

In these situations the random number selection
procedure cannot be used.
© 2009 Thomson South-Western. All Rights Reserved
Slide 9
Simple Random Sampling
From an Infinite Population
 In the case of infinite populations, it is impossible to
obtain a list of all elements in the 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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 10
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
mean .
x as the point estimator of the population
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion p.
© 2009 Thomson South-Western. All Rights Reserved
Slide 11
Example: St. Andrew’s
St. Andrew’s College receives 900 applications
annually 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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 12
Example: St. Andrew’s
The director of admissions would like to know the
following information:
• the average SAT score for the 900 applicants, and
• the proportion of applicants that want to live on
campus.
© 2009 Thomson South-Western. All Rights Reserved
Slide 13
Example: St. Andrew’s
We will now look at two alternatives for obtaining
the desired information.
 Conducting a census of the entire 900 applicants

Selecting a sample of 30 applicants, using Excel
© 2009 Thomson South-Western. All Rights Reserved
Slide 14
Conducting a Census

If the relevant data for the entire 900 applicants were
in the college’s database, the population parameters of
interest could be calculated using the formulas
presented in Chapter 3.

We will assume for the moment that conducting a
census is practical in this example.
© 2009 Thomson South-Western. All Rights Reserved
Slide 15
Conducting a Census


Population Mean SAT Score
xi


 990
900
Population Standard Deviation for SAT Score


2
(
x


)
 i
 80
900
Population Proportion Wanting On-Campus Housing
648
p
 .72
900
© 2009 Thomson South-Western. All Rights Reserved
Slide 16
Simple Random Sampling
 Now suppose that the necessary data on the
current year’s applicants were not yet entered in the
college’s database.
 Furthermore, the Director of Admissions must obtain
estimates of the population parameters of interest for
a meeting taking place in a few hours.
 She decides a sample of 30 applicants will be used.
 The applicants were numbered, from 1 to 900, as
their applications arrived.
© 2009 Thomson South-Western. All Rights Reserved
Slide 17
Simple Random Sampling:
Using a Random Number Table

Taking a Sample of 30 Applicants
• Because the finite population has 900 elements, we
will need 3-digit random numbers to randomly
select applicants numbered from 1 to 900.
•
We will use the last three digits of the 5-digit
random numbers in the third column of the
textbook’s random number table, and continue
into the fourth column as needed.
© 2009 Thomson South-Western. All Rights Reserved
Slide 18
Simple Random Sampling:
Using a Random Number Table

Taking a Sample of 30 Applicants
•
•
•
The numbers we draw will be the numbers of the
applicants we will sample unless
• the random number is greater than 900 or
• the random number has already been used.
We will continue to draw random numbers until
we have selected 30 applicants for our sample.
(We will go through all of column 3 and part of
column 4 of the random number table, encountering
in the process five numbers greater than 900 and
one duplicate, 835.)
© 2009 Thomson South-Western. All Rights Reserved
Slide 19
Simple Random Sampling:
Using a Random Number Table

Use of Random Numbers for Sampling
3-Digit
Applicant
Random Number Included in Sample
744
No. 744
436
No. 436
865
No. 865
790
No. 790
835
No. 835
902
Number exceeds 900
190
No. 190
836
No. 836
. . . and so on
© 2009 Thomson South-Western. All Rights Reserved
Slide 20
Simple Random Sampling:
Using a Random Number Table

Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
.
.
30
498
Applicant
Conrad Harris
Enrique Romero
Fabian Avante
Lucila Cruz
Chan Chiang
.
.
SAT
Score
1025
950
1090
1120
930
.
.
Live OnCampus
Yes
Yes
No
Yes
No
.
.
Emily Morse
1010
No
© 2009 Thomson South-Western. All Rights Reserved
Slide 21
Simple Random Sampling:
Using a Computer

Taking a Sample of 30 Applicants
• Computers can be used to generate random
numbers for selecting random samples.
• For example, Excel’s function
= RANDBETWEEN(1,900)
can be used to generate random numbers between
1 and 900.
• Then we choose the 30 applicants corresponding
to the 30 smallest random numbers as our sample.
© 2009 Thomson South-Western. All Rights Reserved
Slide 22
Point Estimation

x as Point Estimator of 
x

x
n


29, 910
 997
30
s as Point Estimator of 
s

i
 (x
i
 x )2
n1

163, 996
 75.2
29
p as Point Estimator of p
p  20 30  .68
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
© 2009 Thomson South-Western. All Rights Reserved
Slide 23
Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
 = Population mean
990
x = Sample mean
997
 = Population std.
80
s = Sample std.
deviation for
SAT score
75.2
p = Population proportion wanting
campus housing
.72
p = Sample pro-
.68
SAT score
deviation for
SAT score
Point
Estimator
Point
Estimate
SAT score
portion wanting
campus housing
© 2009 Thomson South-Western. All Rights Reserved
Slide 24
Sampling Distribution of x

Process of Statistical Inference
Population
with mean
=?
The value of x 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 x.
© 2009 Thomson South-Western. All Rights Reserved
Slide 25
Sampling Distribution of x
The sampling distribution of x is the probability
distribution of all possible values of the sample
mean x.
Expected Value of
x
E( x ) = 
where:
 = the population mean
© 2009 Thomson South-Western. All Rights Reserved
Slide 26
Sampling Distribution of x

Standard Deviation of x
x 
Standard Deviation of x
for a Finite Population
 N n
x  ( )
n N 1
n
• A finite population is treated as being
infinite if n/N < .05.
• ( N  n) / ( N  1) is the finite correction factor.
•  x is referred to as the standard error of the
mean.
© 2009 Thomson South-Western. All Rights Reserved
Slide 27
Form of the Sampling Distribution of x
When the population has a normal distribution, the
sampling distribution of x is normally distributed
for any sample size.
In most applications, the sampling distribution of x
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.
© 2009 Thomson South-Western. All Rights Reserved
Slide 28
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
x 

80

 14.6
n
30
E( x )  990
© 2009 Thomson South-Western. All Rights Reserved
x
Slide 29
Sampling Distribution of x for SAT Scores
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 x will be
between 980 and 1000?
© 2009 Thomson South-Western. All Rights Reserved
Slide 30
Sampling Distribution of x for SAT Scores
Step 1: Calculate the z-value at the upper endpoint of
the interval.
z = (1000 - 990)/14.6= .68
Step 2: Find the area under the curve to the left of the
upper endpoint.
P(z < .68) = .7517
© 2009 Thomson South-Western. All Rights Reserved
Slide 31
Sampling Distribution of x for SAT Scores
Cumulative Probabilities for
the Standard Normal Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6
.7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7
.7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
.8
.7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9
.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
.
.
.
.
.
.
.
.
© 2009 Thomson South-Western. All Rights Reserved
.
.
.
Slide 32
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .7517
x
990 1000
© 2009 Thomson South-Western. All Rights Reserved
Slide 33
Sampling Distribution of x for SAT Scores
Step 3: Calculate the z-value at the lower endpoint of
the interval.
z = (980 - 990)/14.6= - .68
Step 4: Find the area under the curve to the left of the
lower endpoint.
P(z < -.68) = .2483
© 2009 Thomson South-Western. All Rights Reserved
Slide 34
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .2483
x
980 990
© 2009 Thomson South-Western. All Rights Reserved
Slide 35
Sampling Distribution of x for SAT Scores
Step 5: Calculate the area under the curve between
the lower and upper endpoints of the interval.
P(-.68 < z < .68) = P(z < .68) - P(z < -.68)
= .7517 - .2483
= .5034
The probability that the sample mean SAT score will
be between 980 and 1000 is:
P(980 <
x < 1000) = .5034
© 2009 Thomson South-Western. All Rights Reserved
Slide 36
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .5034
980 990 1000
© 2009 Thomson South-Western. All Rights Reserved
x
Slide 37
Relationship Between the Sample Size
and the Sampling Distribution of x
 Suppose we select a simple random sample of 100
applicants instead of the 30 originally considered.
 E(x ) =  regardless of the sample size. In our
example, E( x) remains at 990.
 Whenever the sample size is increased, the standard
error of the mean  x is decreased. With the increase
in the sample size to n = 100, the standard error of the
mean is decreased to:

80
x 

 8.0
n
100
© 2009 Thomson South-Western. All Rights Reserved
Slide 38
Relationship Between the Sample Size
and the Sampling Distribution of x
With n = 100,
x  8
With n = 30,
 x  14.6
E( x )  990
© 2009 Thomson South-Western. All Rights Reserved
x
Slide 39
Relationship Between the Sample Size
and the Sampling Distribution of x
 Recall that when n = 30, P(980 <
x < 1000) = .5034.
 We follow the same steps to solve for P(980 < x < 1000)
when n = 100 as we showed earlier when n = 30.
 Now, with n = 100, P(980 <
x < 1000) = .7888.
 Because the sampling distribution with n = 100 has a
smaller standard error, the values of x have less
variability and tend to be closer to the population
mean than the values of x with n = 30.
© 2009 Thomson South-Western. All Rights Reserved
Slide 40
Relationship Between the Sample Size
and the Sampling Distribution of x
Sampling
Distribution
of x
x  8
Area = .7888
x
980 990 1000
© 2009 Thomson South-Western. All Rights Reserved
Slide 41
End of Chapter 7, Part A
© 2009 Thomson South-Western. All Rights Reserved
Slide 42