1. Sampling and Sampling Distribution
Statistics
Sampling and Sampling Distribution

2. STATISTICS in PRACTICE
MeadWestvaco Corporation’s
products include textbook
paper, magazine paper, and
office products.
MeadWestvaco’s internal consulting
group uses sampling to provide information that enables the company to obtain significant productivity benefits and remain competitive.

3. STATISTICS in PRACTICE
Managers need reliable and accurate information about the timberlands and forests to evaluate the company’s ability to meet its future raw material needs.
Data collected from sample plots throughout the forests are the basis for learning about the population of trees owned by the company.

4. Contents The Electronics Associates Sampling Problem
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of p
Properties of Point Estimators
Other Sampling Methods

5. Statistical Inference
The purpose of statistical inference is to obtain
information about a population from
information contained in a sample.
A population is the set of all the elements of
interest.
A sample is a subset of the population.

6. 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.

7. The Electronics Associates Sampling Problem
Often the cost of collecting information from a sample is substantially less than from a population,
Especially when personal interviews must be conducted to collect the information.

8. Simple Random Sampling: Finite Population
Finite populations are often defined by lists such as:
Organization membership roster
Credit card account numbers
Inventory product numbers

9. Simple Random Sampling: 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.

10. 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.

11. Simple Random Sampling
Random Numbers: the numbers in the table are random, these four-digit numbers are equally likely.

12. Simple Random Sampling: Infinite Population
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.

13. Simple Random Sampling: Infinite 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.

14. 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.

15. 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 as the point estimator of the
population mean .

16. Point Estimation s is the point estimator of the population standard
deviation .
is the point estimator of the population
proportion p.

17. Point Estimation
Example: to estimate the population mean, the population standard deviation and population proportion.

18. 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.

19. 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.

20. Sampling Error The sampling errors are: for sample mean
for sample standard deviation
for sample proportion

21. 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.

22. 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.

23. Example: St. Andrew’s We will now look at three
alternatives for obtaining
The desired information.
Conducting a census of
the entire 900 applicants
Selecting a sample of 30
applicants, using a random number table
Selecting a sample of 30 applicants, using Excel

24. 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.

25. Conducting a Census Population Mean SAT Score
Population Standard Deviation for SAT Score
Population Proportion Wanting On-Campus Housing

26. 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.

27. 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.

28. Simple Random Sampling
The applicants were numbered, from 1 to 900, as their applications arrived.

29. 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.

30. 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.

31. Simple Random Sampling: Using a Random Number Table
(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.)

32. Simple Random Sampling: Using a Random Number Table
Use of Random Numbers for Sampling
3-Digit
Random Number
Applicant
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

33. Simple Random Sampling: Using a Random Number Table
Sample Data
Random
Number
SAT
Score
Live On-
Campus
No.
Applicant
Conrad Harris Yes
Enrique Romero Yes
Fabian Avante No
Lucila Cruz Yes
Chan Chiang No
Emily Morse No

34. 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.

35. Point Estimation as Point Estimator of  s as Point Estimator of  –
p as Point Estimator of p

36. Point Estimation
Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

37. Summary of Point Estimates Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
Point
Estimator
Point
Estimate
m = Population mean
SAT score
990
= Sample mean
SAT score
997
s = Population std.
deviation for
SAT score 80
s = Sample std.
deviation for
SAT score
75.2
p = Population pro-
portion wanting
campus housing
.72
= Sample pro-
portion wanting
campus housing
.68

38. Sampling Distribution
Example: Relative Frequency Histogram of Sample
Mean Values from 500 Simple Random Samples of
30 each.

39. Sampling Distribution
Example: Relative Frequency Histogram of Sample
Proportion Values from 500 Simple Random
Samples of 30 each.

40. Sampling Distribution of
Process of Statistical Inference
Population
with mean
m = ?
A simple random sample
of n elements is selected
from the population.
The value of is used to
make inferences about
the value of m.
The sample data
provide a value for
the sample mean .

41. Sampling Distribution of
The sampling distribution of is the probability
distribution of all possible values of the sample
mean .
Expected Value of
E( ) = 
where:
 = the population mean

42. Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population
A finite population is treated as being
infinite if n/N < .05.

43. Sampling Distribution of
is the finite correction factor.
is referred to as the standard error of the mean.

44. Form of the Sampling Distribution of
If we use a large (n > 30) simple random sample,
the central limit theorem enables us to conclude
that the sampling distribution of can be
approximated by a normal distribution.
When the simple random sample is small (n < 30),
the sampling distribution of can be considered
normal only if we assume the population has a
normal distribution.

45. Central Limit Theorem
Illustration of The Central Limit Theorem

46. Relationship Between the Sample Size and the Sampling Distribution of Sample Mean
A Comparison of The Sampling Distributions of Sample Mean for Simple Random Samples of n = 30 and n = 100.

47. Sampling Distribution of for SAT Scores

48. Sampling Distribution of 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
will be between 980 and 1000?

49. Sampling Distribution of for SAT Scores
Step 1: Calculate the z-value at the upper
endpoint of the interval.
z = ( )/14.6= .68
Step 2: Find the area under the curve to the
left of the upper endpoint.
P(z < .68) = .7517

50. Sampling Distribution of for SAT Scores
Cumulative Probabilities for
the Standard Normal Distribution

51. Sampling Distribution of for SAT Scores
Area = .7517
990
1000

52. Sampling Distribution of for SAT Scores
Step 3: Calculate the z-value at the lower
endpoint of the interval.
z = ( )/14.6= - .68
Step 4: Find the area under the curve to the
left of the lower endpoint.
P(z < -.68) = P(z > .68)
= .2483 =
= 1 - P(z < .68)

53. Sampling Distribution of for SAT Scores
Area = .2483
980
990

54. Sampling Distribution of 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) =
= .5034
The probability that the sample mean SAT score will be between 980 and 1000 is:
P(980 < < 1000) = .5034

55. Sampling Distribution of for SAT Scores
Area = .5034
980
990
1000

56. Relationship Between the Sample Size and the Sampling Distribution of
Suppose we select a simple random sample
of 100 applicants instead of the 30 originally
considered.
E( ) = m regardless of the sample size.
in our example, E( ) remains at 990.

57. Relationship Between the Sample Size and the Sampling Distribution of
Whenever the sample size is increased, the
standard error of the mean is decreased.
With the increase in the sample size to n = 100,
the standard error of the mean is decreased to:

58. Relationship Between the Sample Size and the Sampling Distribution of
With n = 100,
With n = 30,

59. Relationship Between the Sample Size and the Sampling Distribution of
Recall that when n = 30,
P(980 < < 1000) =
We follow the same steps to solve for
P(980 < < 1000)
when n = 100 as we showed earlier when n = 30.

60. Relationship Between the Sample Size and the Sampling Distribution of
Now, with n = 100, P(980 < < 1000) =
Because the sampling distribution with n = 100
has a smaller standard error, the values of have
less variability and tend to be closer to the
population mean than the values of with n = 30.

61. Relationship Between the Sample Size and the Sampling Distribution of
Area = .7888
980
990
1000

62. Sampling Distribution
Example: Relative Frequency Histogram of Sample Proportion Values from 500 Simple Random Samples of 30 each.

63. of n elements is selected
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 value of is used
to make inferences
about the value of p.
The sample data
provide a value for the
sample proportion .

64. Sampling Distribution of
The sampling distribution of p is the probability
distribution of all possible values of the sample
proportion p .
Expected Value of p
where:
p = the population proportion

65. Sampling Distribution of
Standard Deviation of p
Finite Population
Infinite Population
is referred to as the standard error of the proportion.

66. Form of the Sampling Distribution of
The sampling distribution of 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

67. Form of the Sampling Distribution of
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.

68. Sampling Distribution of
Example: St. Andrew’s College
Recall that 72% of the prospective students
applying to St. Andrew’s College desire
on-campus housing.

69. Sampling Distribution of
Example: St. Andrew’s College
What is the probability that a simple random
sample of 30 applicants will provide an estimate
of the population proportion of applicant
desiring on-campus housing that is within plus or
minus .05 of the actual population proportion?

70. Sampling Distribution of
For our example, with n = 30 and p = .72,
the normal distribution is an acceptable
approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 - p) = 30(.28) = 8.4 > 5

71. Sampling Distribution of

72. Sampling Distribution of
Step 1: Calculate the z-value at the upper
endpoint of the interval.
z = ( ) /.082 = .61
Step 2: Find the area under the curve to the
left of the upper endpoint.
P(z < .61) = .7291

73. Sampling Distribution of
Cumulative Probabilities for
the Standard Normal Distribution

74. Sampling Distribution of
Area = .7291
.72
.77

75. Sampling Distribution of
Step 3: Calculate the z-value at the lower
endpoint of the interval.
z = ( ) /.082 = - .61
Step 4: Find the area under the curve to the
left of the lower endpoint.
P(z < -.61) = P(z > .61)
= 1 - P(z < .61) =
= .2709

76. Sampling Distribution of
Area = .2709
.67
.72

77. Sampling Distribution of
Step 5: Calculate the area under the curve between
the lower and upper endpoints of the interval.
P(-.61 < z < .61) = P(z < .61) - P(z < -.61) =
= .4582
The probability that the sample proportion of
applicants wanting on-campus housing will be
within +/-.05 of the actual population proportion :
P(.67 < < .77) = .4582

78. Sampling Distribution of
Area = .4582
.67
.72
.77

79. Point Estimators Notations: θ = the population parameter of interest.
For example, population mean, population standard deviation, population proportion, and so on. ^

80. Point Estimators Notations:
θ = the sample statistic or point estimator of θ .
Represents the corresponding sample statistic such as the sample mean, sample standard deviation, and sample proportion.
The notation θ is the Greek letter theta.
the notation θ is pronounced “theta-hat.” ^ ^

81. 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

82. Properties of Point Estimators
Unbised
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.

83. Properties of Point Estimators
Unbised
The sample statistic θ is unbiased estimator of the population parameter θ if ^ ^
E(θ)= θ
where ^
E(θ)=the expected value of the sample statistic θ

84. Properties of Point Estimators
Examples of Unbiased and Biased Point Estimators

85. 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.

86. Properties of Point Estimators
Example: Sampling Distributions of Two
Unbiased Point Estimators.

87. 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.

88. Other Sampling Methods
Stratified Random Sampling(分層隨機抽樣)
Cluster Sampling(部落抽樣）
Systematic Sampling（系統抽樣）
Convenience Sampling（便利抽樣）
Judgment Sampling(判斷抽樣）

89. 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).

90. Stratified Random Sampling
Diagram for Stratified Random Sampling

91. 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.

92. Stratified Random Sampling
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.

93. 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.

94. Cluster Sampling
Diagram for Cluster Sampling

95. 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.

96. 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.

97. 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

98. 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.

99. 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.

100. 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.

101. 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.