Download 1 Chapter 7 Sampling and Sampling Distributions Learning

Chapter 7
Sampling and Sampling Distributions
Slide 1
Learning objectives
1. Understand Simple Random Sampling
2. Understand Point Estimation and be able to
compute point estimates
3. Understand Sampling Distribution of x
4. Understand Sampling Distribution of p
5. Understand properties of Point Estimators
6. Understand other Sampling Methods
Slide 2
1
Statistical Inference
A primary purpose of statistical inference is to
develop estimates and test hypotheses about
population parameters using information contained
in a sample.
A parameter is a numerical characteristic of a
population. (e.g., mean and variance)
With proper sampling methods, the sample results
can provide “good” estimates of the population
characteristics.
Slide 3
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Simple Random Sampling
Simple Random Sampling
Finite population (N)
Infinite population
Probability of selecting any
one data point = 1/N
N is unknown.
Without replacement
With replacement
1. From the population
2. Selected independently
Slide 4
2
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
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 x as the point estimator of the population
mean µ.
s is the point estimator of the population standard
deviation σ.
p is the point estimator of the population proportion p.
Slide 5
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Error
n We cannot expect sample statistics to exactly equal
the parameters
n The absolute value of the difference between an
unbiased point estimate and the corresponding
population parameter is called the sampling error.
(e.g., | x − µ | )
n Sampling error is the result of using a subset of
the population (the sample), and not the entire
population.
n Statistical methods can be used to make
probability statements about the size of the
sampling error.
Slide 6
3
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
n
Sampling Distribution of x
Process of Statistical Inference
A simple random sample
of n elements is selected
from the population.
Population
with mean
µ=?
The value of x is used to
make inferences about
the value of µ.
The sample data
provide a value for
the sample mean x .
Slide 7
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
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
Slide 8
4
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of x
Standard Deviation of x
Finite Population
σx = (
σ N −n
)
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.
Slide 9
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
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.
Figure 7.3 on page 273
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.
Slide 10
5
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of x
for EAI example (section 7.1)
Sampling
Distribution
of x
σx =
σ
4000
=
= 730.3
n
30
E( x ) = $51,800
x
Slide 11
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of x
for EAI example (section 7.1)
What is the probability that a simple random sample
of 30 EAI managers will provide an estimate of the
population mean annual salary that is within +/− $500
of the actual population mean µ ?
In other words, what is the probability that x will be
between 52,300 and 51,300?
Slide 12
6
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
§
Practical value of sample distribution
EAI example
Sampling
Distribution
of x
σ x = 730.3
Area = .5034
51,300 51,800 52,300
x
Slide 13
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Relationship Between the Sample Size
and the Sampling Distribution of x
n Suppose we select a simple random sample of 100
managers instead of the 30 originally considered.
n E(x) = µ regardless of the sample size. In our
example, E(x) remains at 51,800.
n Whenever the sample size is increased, the standard
error of the mean σ xx is decreased. With the increase
in the sample size to n = 100, the standard error of the
mean is decreased to:
σx =
σ
4000
=
= 400
n
100
Slide 14
7
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Relationship Between the Sample Size
and the Sampling Distribution of x
With n = 100,
σx = 400
With n = 30,
σx =730.3
E(x ) = 51,800
x
Slide 15
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
n
n
In-class exercise
#18 (p.277)
#19 (p.277)
Slide 16
8
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
n
Sampling Distribution of p
Making Inferences about a Population Proportion
Population
with proportion
p=?
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for the
sample proportion p.
The value of p is used
to make inferences
about the value of p.
Slide 17
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of p
The sampling distribution of p is the probability
distribution of all possible values of the sample
proportion p .
Expected Value of p
E( p) = p
where:
p = the population proportion
Slide 18
9
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of p
Standard Deviation of p
Finite Population
σp =
Infinite Population
p (1 − p ) N − n
n
N −1
σp =
p (1 − p )
n
σ p is referred to as the standard error of the
proportion.
Slide 19
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Form of the Sampling Distribution of p
The sampling distribution of p can be approximated
by a normal distribution whenever the sample size
is large.
The sample size is considered large whenever these
conditions are satisfied:
np > 5
and
n(1 – p) > 5
Slide 20
10
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Form of the Sampling Distribution of p
For values of p near .50, sample sizes as small as 10
permit a normal approximation.
With very small (approaching 0) or very large
(approaching 1) values of p, much larger samples
are needed.
Slide 21
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Practical value of Sampling Distribution of p
n
What is the probability that a simple random sample
of 30 applicants will provide an estimate of the
population proportion of managers who participated
in the training program within plus or minus .05 of
the actual population proportion?
n
That is, what is the probability of obtaining a sample
with a sample proportion p between .55 and .65
Slide 22
11
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Sampling Distribution of p
Sampling
Distribution
of p
σ p = .0894
Area = .4246
p
.55
.60
.65
Slide 23
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
n
n
In-class exercise
#31 (p.283)
#32 (p.283)
Slide 24
12
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
n
Properties of Point Estimators
Before using a sample statistic as a point estimator,
statisticians check to see whether the sample statistic
has the following properties associated with good
point estimators.
Unbiased
Efficiency
Consistency
Slide 25
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Properties of Point Estimators
Unbiased
If the expected value of the sample statistic is
equal to the population parameter being estimated,
the sample statistic is said to be an unbiased
estimator of the population parameter.
Slide 26
13
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Properties of Point Estimators
Efficiency
Given the choice of two unbiased estimators of
the same population parameter, we would prefer to
use the point estimator with the smaller standard
deviation, since it tends to provide estimates closer to
the population parameter.
The point estimator with the smaller standard
deviation is said to have greater relative efficiency
than the other.
Slide 27
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Properties of Point Estimators
Consistency
A point estimator is consistent if the values of the
point estimator tend to become closer to the
population parameter as the sample size becomes
larger.
Slide 28
14
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Other Sampling Methods
n
Stratified Random Sampling
n
Cluster Sampling
n
Systematic Sampling
n
Convenience Sampling
n
Judgment Sampling
Slide 29
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Stratified Random Sampling
The population is first divided into groups of
elements called strata.
Each element in the population belongs to one and
only one stratum.
Best results are obtained when the elements within
each stratum are as much alike as possible
(i.e. a homogeneous group).
Slide 30
15
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Stratified Random Sampling
A simple random sample is taken from each stratum.
Formulas are available for combining the stratum
sample results into one population parameter
estimate.
Advantage: If strata are homogeneous, this method
is as “precise” as simple random sampling but with
a smaller total sample size.
Example: The basis for forming the strata might be
department, location, age, industry type, and so on.
Slide 31
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Cluster Sampling
The population is first divided into separate groups
of elements called clusters.
Ideally, each cluster is a representative small-scale
version of the population (i.e. heterogeneous group).
A simple random sample of the clusters is then taken.
All elements within each sampled (chosen) cluster
form the sample.
Slide 32
16
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Cluster Sampling
Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.
Advantage: The close proximity of elements can be
cost effective (i.e. many sample observations can be
obtained in a short time).
Disadvantage: This method generally requires a
larger total sample size than simple or stratified
random sampling.
Slide 33
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Systematic Sampling
If a sample size of n is desired from a population
containing N elements, we might sample one
element for every n/N elements in the population.
We randomly select one of the first n/N elements
from the population list.
We then select every n/Nth element that follows in
the population list.
Slide 34
17
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Systematic Sampling
This method has the properties of a simple random
sample, especially if the list of the population
elements is a random ordering.
Advantage: The sample usually will be easier to
identify than it would be if simple random sampling
were used.
Example: Selecting every 100th listing in a telephone
book after the first randomly selected listing
Slide 35
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Convenience Sampling
It is a nonprobability sampling technique. Items are
included in the sample without known probabilities
of being selected.
The sample is identified primarily by convenience.
Example: A professor conducting research might use
student volunteers to constitute a sample.
Slide 36
18
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Convenience Sampling
Advantage: Sample selection and data collection are
relatively easy.
Disadvantage: It is impossible to determine how
representative of the population the sample is.
Slide 37
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Judgment Sampling
The person most knowledgeable on the subject of the
study selects elements of the population that he or
she feels are most representative of the population.
It is a nonprobability sampling technique.
Example: A reporter might sample three or four
senators, judging them as reflecting the general
opinion of the senate.
Slide 38
19
•L.O.
•Simple random sampling
•Point estimation
•Sampling distribution of x
•Sampling distribution of p
•Properties of point estimators
•Other sampling methods
Judgment Sampling
Advantage: It is a relatively easy way of selecting a
sample.
Disadvantage: The quality of the sample results
depends on the judgment of the person selecting the
sample.
Slide 39
End of Chapter 7
Slide 40
20