Download Lecture 5 Point estimation and sampling distributions

LECTURE 5
POINT
E S T I M AT I O N
AND SAMPLING
DISTRIBUTION
S
Dr. Wendy Jiao
[email protected]
AGENDA

Simple Random Sampling

Point Estimation

Sampling Distribution of 𝑥

Sampling Methods
2
P O P U L AT I O N S A N D
SAMPLES

Population – is a collection of all the
elements of interest.

Sample – portion/subset of the
population selected for the purpose of
the study

Population parameters – numerical
characteristics of interest in a study
(won’t know value with certainty if using
a sample)

Sample statistics – characteristics of
interest derived from a sample (values
will vary from one sample to the next)

Two questions to think about:
1.
2.
Is a full population necessary to get your
desired info?
How should a sample be selected – are
some methods/samples better than
others?
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SAMPLING FROM A FINITE POPULATION
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.
Replacing each sampled
element before selecting
subsequent elements is
called sampling with
replacement. An element
can appear in the sample
more than once.
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.
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SAMPLING FROM A FINITE POPULATION
EXAMPLE: ST. ANDREW’S
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.
Step 1:
Assign a random number to each of the 900 applicants.
Step 2:
Select the 30 applicants corresponding to the 30 smallest
random numbers.
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SAMPLING FROM AN INFINITE
P O P U L AT I O N
 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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SAMPLING FROM AN INFINITE
P O P U L AT I O N
 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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AGENDA

Simple Random Sampling

Point Estimation

Sampling Distribution of 𝑥

Sampling Methods
8
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.
Sample mean X is the point estimator of the population
mean .
Sample standard deviation S is the point estimator of
the population standard deviation .
Sample proportion P is the point estimator of the
population proportion p.
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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.
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Sampling Error

The sampling errors are:
|x   | for sample mean
|s   | for sample standard deviation
| p  p | for sample proportion
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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.
The director of admissions would like to know the following
information: the average SAT score for the 900 applicants, and the
proportion of applicants who want to live on campus. We shall
now look at two alternatives for obtaining the desired information.
• Conducting a census of the entire 900 applicants.
• Selecting a sample of 30 applicants.
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Conducting a Census

Population Mean SAT Score:
xi


 990
900

Population Standard Deviation for SAT Score:


2
(
x


)
 i
900
 80
Population Proportion wanting On-Campus Housing:
648
p
 0.72
900
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Simple Random Sampling:
Using a Random Number Table

Taking a Sample of 30 Applicants
• Because the finite population has 900 elements, we
shall need 3-digit random numbers to randomly
select applicants numbered from 1 to 900.
•
•
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 shall continue to draw random numbers until
we have selected 30 applicants for our sample.
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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
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Point Estimation

x as Point Estimate of 
x

x

29, 910

 997
n
30
s as Point Estimate of 
s

i
2
(
x

x
)
 i
n1
p as Point Estimate of p
163, 996

 75.2
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p  20 30  0.68
Note: Different random numbers would have identified a different
sample which would have resulted in different point estimates.
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Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
 = Population mean
990
X = Sample mean
 = Population std.
80
S = Sample std.
deviation for
SAT score
p = Population pro-
0.72
SAT score
deviation for
SAT score
portion wanting
campus housing
Point
Estimator
SAT score
Point
Estimate
997
75.2
0.68
P = Sample proportion wanting
campus housing
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AGENDA

Simple Random Sampling

Point Estimation

Sampling Distribution of 𝑥

Sampling Methods
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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.
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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
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Sampling Distribution of X
Standard Deviation of X
Finite Population, if

N n
X  ( )
n N 1
𝑛
𝑁
> 0.05
Infinite Population, if
X 
𝑛
𝑁
≤ 0.05

n
• A finite population is treated as being infinite if n/N < 0.05.
• ( N  n) / ( N  1) is the finite population correction factor.
•  X is referred to as the standard error of the mean.
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Central limit theorem (CLT)

1)
2)
The CLT is very important in statistics and has several important consequences.
Amongst those are these two facts:
When we add n random variables together, regardless of the distribution of the
individual addends, the distribution of the sum tends towards a normal
distribution as n tends to infinity.
• We saw this the R tutorial when we looked at the histogram of rolling m dice.
When we sample with a fixed sample size n repetitively from a population with
mean 𝜇 and standard deviation 𝜎, the means of the sample tend towards a normal
𝜎
distribution with a mean of 𝜇, and a standard deviation .
𝑛
• We will use this in this week lecture.

Usually we do not have to let n = infinity to make good use of these facts. In many
practical situations, n>30 is sufficient to give reliable results.
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Form of the Sampling Distribution of X
When the population has a normal distribution, the
sampling distribution of X is normal in shape 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.
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Sampling Distribution of X for SAT Scores
Sampling
Distribution
of X
E( X )  990
X 

80

 14.6
n
30
x
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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?
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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= 0.68
Step 2: Find the area under the curve to the left of the
upper endpoint.
P(Z < 0.68) = 0.7517
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Sampling Distribution of X for SAT Scores
Sampling
Distribution
of X
 X  14.6
Area = 0.7517
x
990 1000
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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= - 0.68
Step 4: Find the area under the curve to the left of the
lower endpoint.
P(Z < -0.68) = 0.2483
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Sampling Distribution of X for SAT Scores
Sampling
Distribution
of X
 X  14.6
Area = 0.2483
x
980 990
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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(-0.68 < Z < 0.68) = P(Z < 0.68) - P(Z < -0.68)
= 0.7517 - 0.2483
= 0.5034
The probability that the sample mean SAT score will
be between 980 and 1000 is:
P(980 < X < 1000) = 0.5034
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Sampling Distribution of X for SAT Scores
Sampling
Distribution
of X
 X  14.6
Area = 0.5034
980 990 1000
x
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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
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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
x
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Relationship Between the Sample Size
and the Sampling Distribution of X
 Recall that when n = 30, P(980 < X < 1000) = 0.5034.
 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) = 0.7888.
 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.
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Relationship Between the Sample Size
and the Sampling Distribution of X
Sampling
Distribution
of X
Sample size n = 100
X  8
Area = 0.7888
x
980 990 1000
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AGENDA

Simple Random Sampling

Point Estimation

Sampling Distribution of 𝑥

Sampling Methods







Fixed interval sampling
Random sampling
Variable attribute sampling
Continuous sampling
Stratified Random Sampling
Cluster Sampling
Convenience Sampling
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F I X E D I N T E RVA L S A M P L I N G
( S Y S T E M AT I C S A M P L I N G )

If a sample size of n is desired from a population
containing N elements, we might sample one
𝑁
element for every 𝑛 elements in the population.

We randomly select one of the first
the population list.

We then select every 𝑛 th element that follows in the
population list.

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 100 th listing in a telephone
book after the first randomly selected listing.
𝑁
𝑛
elements from
𝑁
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F I X E D I N T E RVA L S A M P L I N G C A N
P R O D U C E R E S U LT S W I T H S Y S T E M AT I C E R R O R S …
1. consider the case of determining the utilisation of the Streatham Court C
2. suppose we decide to
take samples at 5 minutes
past the hour…
5 mins past hour
pˆ  16
40  40%
3. suppose we decide to
take samples at 30
minutes past the hour…
30 mins past hour
pˆ  33
40  82.5%
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RANDOM SAMPLING
means samples are taken after random intervals of time.
The length of each interval is driven by a random number drawn
from a uniform distribution.
This method has the advantage that the results obtained are
unbiased.
Biased could be introduced by
the person conducting the activity sampling study,
the subjects (operators / workers) of the study,
systematic structure in the process being studied.
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RANDOM SAMPLING
Random intervals between samples mean that

samples are not taken at regular intervals
- operators will not be able to predict when the next sample will be taken

samples will not be taken when it is convenient for the sampler

the impact of any systematic effects will be reduced
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VA R I A B L E AT T R I B U T E
SAMPLING
This is when we sample at different
frequencies at different times
a customer arrives at the
process
For example we may
wish to sample when
quality problems increase
peak demands occur (at lunch
time or the weekends for
example)
the sun shines or when it rains
Difficult to be sure whether the
attribute determining when samples
are taken is relevant
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CONTINUOUS SAMPLING

This is the only truly accurate
sampling strategy (but it is NOT a
true activity sampling strategy).

Continuous sampling is expensive
and time consuming.
 However, the data collection required
to do this could automated with
technology (such as RFID or bar
codes), or the data could be gathered
from CCTV cameras, or computer
logs to can investigated.
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DIFFERENT SAMPLING
S T R AT E G I E S
S T R AT I F I E D R A N D O M S A M P L I N G

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).

Example: The basis for forming the
strata might be department, location,
age, industry type, and so on.
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S T R AT I F I E D R A N D O M S A M P L I N G

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 provides
results that is as “precise” as simple random sampling but with a
smaller total sample size.
45
CLUSTER SAMPLING (1 OF 2)

The population is first divided into separate
groups of elements called clusters.

Ideally, each cluster is a representative smallscale 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.

In the ideal case, each cluster is a
representative small-scale version of the
entire population.
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CLUSTER SAMPLING
(2
OF
2)

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.
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CONVENIENCE SAMPLING

It is a nonprobability sampling technique.

The sample is identified primarily by convenience.
 Items are chosen simply by virtue of their being convenient for the
analyst to identify and select.

Example: mall-intercept interviews, tear-out survey
questionnaires from magazines, people-on-the-street interviews,
customer-satisfaction surveys.

Advantage: Sample selection and data collection are relatively
easy.

Disadvantage: It is impossible to determine how representative
of the population the sample is.
48
R E C O M M E N D AT I O N

It is recommended that probability sampling methods (simple
random, stratified, cluster, or systematic) be used.

For these methods, formulas are available for evaluating the
“goodness” of the sample results in terms of the closeness of the
results to the population parameters being estimated.

An evaluation of the goodness cannot be made with nonprobability (convenience or judgment) sampling methods.
49