1. Chapter 5, Part A [SBE 7/e, Ch. 7] Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of

2. 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.

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.

4. Simple Random Sampling: 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.

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.

6. 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.
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.

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.

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.
We refer to as the point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
is the point estimator of the population proportion p.

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.

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

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.

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.

13. 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

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.

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

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.

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.

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.)

19. 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

20. 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

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.

22. Point Estimation as Point Estimator of  s as Point Estimator of 
as Point Estimator of p
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.

23. 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

24. 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 .

25. 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

26. Sampling Distribution of
Standard Deviation of
Finite Population
Infinite Population
A finite population is treated as being
infinite if n/N < .05.
is the finite correction factor.
is referred to as the standard error of the
mean.

27. 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.

28. Sampling Distribution of for SAT Scores

29. 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?

30. 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

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

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

33. 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)
= 1 - P(z < .68) =
= .2483

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

35. 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) = =
The probability that the sample mean SAT score will
be between 980 and 1000 is:
P(980 < < 1000) = .5034

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

37. 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.
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:

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

39. 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.
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.

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

41. Chapter 5, Part B [SBE 7/e, Ch. 7] Sampling and Sampling Distributions
Sampling Distribution of
Properties of Point Estimators
Other Sampling Methods

42. Sampling Distribution of
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 .

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

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

45. 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

46. 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.

47. 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.
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?

48. 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

49. Sampling Distribution of

50. 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

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

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

53. 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

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

55. 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) = =
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

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

57. 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

58. 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.

59. 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.

60. 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.

61. Other Sampling Methods
Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling

62. 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).

63. 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.

64. 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.

65. 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.

66. 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.

67. 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

68. 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.

69. 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.

70. 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.

71. 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.