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Slides Prepared by
JOHN S. LOUCKS
St. Edward’s
Edward’s University
© 2006 Thomson/South-Western
Slide 1
Chapter 7, Part A
Sampling and Sampling Distributions
„ Simple Random Sampling
„ Point Estimation
„ Introduction to Sampling Distributions
„ Sampling Distribution of x
© 2006 Thomson/South-Western
Slide 2
Statistical Inference
The
The purpose
purpose of
of statistical
statistical inference
inference is
is to
to obtain
obtain
information
information about
about aa population
population from
from information
information
contained
contained in
in aa sample.
sample.
A
A population
population is
is the
the set
set of
of all
all the
the elements
elements of
of interest.
interest.
A
A sample
sample is
is aa subset
subset of
of the
the population.
population.
© 2006 Thomson/South-Western
Slide 3
Statistical Inference
The
The sample
sample results
results provide
provide only
only estimates
estimates of
of the
the
values
values of
of the
the population
population characteristics.
characteristics.
With
With proper
proper sampling
sampling methods
methods,, the
the sample
sample results
results
can
good” estimates
can provide
provide ““good”
estimates of
of the
the population
population
characteristics.
characteristics.
A
A parameter
parameter is
is aa numerical
numerical characteristic
characteristic of
of aa
population.
population.
© 2006 Thomson/South-Western
Slide 4
Simple Random Sampling:
Finite Population
Q
Finite populations are often defined by lists such as:
• Organization membership roster
• Credit card account numbers
• Inventory product numbers
Q
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.
© 2006 Thomson/South-Western
Slide 5
Simple Random Sampling:
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.
© 2006 Thomson/South-Western
Slide 6
Simple Random Sampling:
Infinite Population
Q
Q
Infinite populations are often defined by an ongoing
process whereby the elements of the population
consist of items generated as though the process would
operate indefinitely.
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.
© 2006 Thomson/South-Western
Slide 7
Simple Random Sampling:
Infinite Population
„ In the case of infinite populations, it is impossible to
obtain a list of all elements in the population.
„ The random number selection procedure cannot be
used for infinite populations.
© 2006 Thomson/South-Western
Slide 8
Point Estimation
In
In point
point estimation
estimation we
we use
use the
the data
data from
from the
the sample
sample
to
to compute
compute aa value
value of
of aa sample
sample statistic
statistic that
that serves
serves
as
as an
an estimate
estimate of
of aa population
population parameter.
parameter.
We
We refer
refer to
to x as
as the
the point
point estimator
estimator of
of the
the population
population
mean
mean μμ..
ss is
is the
the point
point estimator
estimator of
of the
the population
population standard
standard
deviation
deviation σσ..
p is
is the
the point
point estimator
estimator of
of the
the population
population proportion
proportion pp..
© 2006 Thomson/South-Western
Slide 9
Sampling Error
„ When the expected value of a point estimator is equal
to the population parameter, the point estimator is said
to be unbiased.
„ The absolute value of the 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.
„ Statistical methods can be used to make probability
statements about the size of the sampling error.
© 2006 Thomson/South-Western
Slide 10
Sampling Error
Q
The sampling errors are:
|x − μ | for sample mean
|s − σ | for sample standard deviation
| p − p| for sample proportion
© 2006 Thomson/South-Western
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.
© 2006 Thomson/South-Western
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.
© 2006 Thomson/South-Western
Slide 13
Example: St. Andrew’s
We will now look at three
alternatives for obtaining the
desired information.
Q Conducting a census of the
entire 900 applicants
Q Selecting a sample of 30
applicants, using a random number table
Q
Selecting a sample of 30 applicants, using Excel
© 2006 Thomson/South-Western
Slide 14
Conducting a Census
Q
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.
Q
We will assume for the moment that conducting a
census is practical in this example.
© 2006 Thomson/South-Western
Slide 15
Conducting a Census
Q
Q
Population Mean SAT Score
xi
∑
μ=
= 990
900
Population Standard Deviation for SAT Score
σ=
Q
2
(
x
−
μ
)
∑ i
900
= 80
Population Proportion Wanting On-Campus Housing
648
p=
= .72
900
© 2006 Thomson/South-Western
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.
© 2006 Thomson/South-Western
Slide 17
Simple Random Sampling:
Using a Random Number Table
Q
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.
© 2006 Thomson/South-Western
Slide 18
Simple Random Sampling:
Using a Random Number Table
Q
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.)
© 2006 Thomson/South-Western
Slide 19
Simple Random Sampling:
Using a Random Number Table
Q
Use of Random Numbers for Sampling
3-Digit
Applicant
Random Number Included in Sample
No. 744
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
© 2006 Thomson/South-Western
Slide 20
Simple Random Sampling:
Using a Random Number Table
Q
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
© 2006 Thomson/South-Western
Slide 21
Simple Random Sampling:
Using a Computer
Q
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.
© 2006 Thomson/South-Western
Slide 22
Point Estimation
Q
x as Point Estimator of μ
x
∑
x=
29,910
=
= 997
30
30
i
Q
s as Point Estimator of σ
s=
Q
∑ (x
i
− x )2
29
=
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.
© 2006 Thomson/South-Western
Slide 23
Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
μ = Population mean
990
x = Sample mean
997
σ = Population std.
80
s = Sample std.
deviation for
SAT score
75.2
p = Population proportion wanting
campus housing
.72
p = Sample proportion wanting
campus housing
SAT score
deviation for
SAT score
© 2006 Thomson/South-Western
Point
Estimator
Point
Estimate
SAT score
.68
Slide 24
Sampling Distribution of x
Q
Process of Statistical Inference
Population
with mean
μ=?
The value of x is used to
make inferences about
the value of μ.
© 2006 Thomson/South-Western
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x .
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 ) = μ
where:
μ = the population mean
© 2006 Thomson/South-Western
Slide 26
Sampling Distribution of x
Standard Deviation of x
Finite Population
σ
N −n
σx = ( )
n N −1
Infinite Population
σx =
σ
n
• A finite population is treated as being
infinite if n/N < .05.
• ( N − n ) / ( N − 1) is the finite correction factor.
• σ x is referred to as the standard error of the
mean.
© 2006 Thomson/South-Western
Slide 27
Form of the Sampling Distribution of x
If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
sampling distribution of x can be approximated by
a normal distribution.
When the simple random sample is small (n < 30),
the sampling distribution of x can be considered
normal only if we assume the population has a
normal distribution.
© 2006 Thomson/South-Western
Slide 28
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
E( x ) = 990
© 2006 Thomson/South-Western
σx =
σ
n
=
80
= 14.6
30
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 μ ?
In other words, what is the probability that x will be
between 980 and 1000?
© 2006 Thomson/South-Western
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
© 2006 Thomson/South-Western
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
.
.
.
.
.
.
.
.
.
.
.
© 2006 Thomson/South-Western
Slide 32
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
σ x = 14.6
Area = .7517
x
990 1000
© 2006 Thomson/South-Western
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) = P(z > .68)
= 1 - P(z < .68)
= 1 - . 7517
= .2483
© 2006 Thomson/South-Western
Slide 34
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
σ x = 14.6
Area = .2483
x
980 990
© 2006 Thomson/South-Western
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
© 2006 Thomson/South-Western
Slide 36
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
σ x = 14.6
Area = .5034
980 990 1000
© 2006 Thomson/South-Western
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) = μ 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 σ x is decreased. With the increase
in the sample size to n = 100, the standard error of the
mean is decreased to:
σ
80
σx =
=
= 8.0
100
n
© 2006 Thomson/South-Western
Slide 38
Relationship Between the Sample Size
and the Sampling Distribution of x
With n = 100,
σx = 8
With n = 30,
σx = 14.6
E( x ) = 990
© 2006 Thomson/South-Western
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.
© 2006 Thomson/South-Western
Slide 40
Relationship Between the Sample Size
and the Sampling Distribution of x
Sampling
Distribution
of x
σx = 8
Area = .7888
x
980 990 1000
© 2006 Thomson/South-Western
Slide 41
End of Chapter 7, Part A
© 2006 Thomson/South-Western
Slide 42