point estimation

1. 1Slide
Chapter 7
Sampling and Sampling Distributions
n Simple Random Sampling
n Point Estimation
n Introduction to Sampling Distributions
n Sampling Distribution of
n Sampling Distribution of
n Properties of Point Estimators
n Other Sampling Methods
x
p n = 100
n = 30

2. 2Slide
Statistical Inference
n The purpose of statistical inference is to obtain
information about a population from information
contained in a sample.
n A population is the set of all the elements of interest.
n A sample is a subset of the population.
n The sample results provide only estimates of the
values of the population characteristics.
n A parameter is a numerical characteristic of a
population.
n With proper sampling methods, the sample results
will provide “good” estimates of the population
characteristics.

3. 3Slide
Simple Random Sampling
n Finite Population
•A simple random sample 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.
•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.

4. 4Slide
n 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.
•The population is usually considered infinite if it
involves an ongoing process that makes listing or
counting every element impossible.
•The random number selection procedure cannot
be used for infinite populations.
Simple Random Sampling

5. 5Slide
Other Sampling Methods
n Stratified Random Sampling
n Cluster Sampling
n Systematic Sampling
n Convenience Sampling
n Judgment Sampling

6. 6Slide
Stratified Random Sampling
n The population is first divided into groups of
elements called strata.
n Each element in the population belongs to one and
only one stratum.
n Best results are obtained when the elements within
each stratum are as much alike as possible (i.e.
homogeneous group).
n A simple random sample is taken from each stratum.
n Formulas are available for combining the stratum
sample results into one population parameter
estimate.

7. 7Slide
Stratified Random Sampling
n Advantage: If strata are homogeneous, this method
is as “precise” as simple random sampling but with a
smaller total sample size.
n Example: The basis for forming the strata might be
department, location, age, industry type, etc.

8. 8Slide
Cluster Sampling
n The population is first divided into separate groups
of elements called clusters.
n Ideally, each cluster is a representative small-scale
version of the population (i.e. heterogeneous group).
n A simple random sample of the clusters is then taken.
n All elements within each sampled (chosen) cluster
form the sample.
… continued

9. 9Slide
Cluster Sampling
n Advantage: The close proximity of elements can be
cost effective (I.e. many sample observations can be
obtained in a short time).
n Disadvantage: This method generally requires a
larger total sample size than simple or stratified
random sampling.
n Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.

10. 10Slide
Systematic Sampling
n 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.
n We randomly select one of the first n/N elements
from the population list.
n We then select every n/Nth element that follows in
the population list.
n This method has the properties of a simple random
sample, especially if the list of the population
elements is a random ordering.
… continued

11. 11Slide
Systematic Sampling
n Advantage: The sample usually will be easier to
identify than it would be if simple random sampling
were used.
n Example: Selecting every 100th listing in a telephone
book after the first randomly selected listing.

12. 12Slide
Convenience Sampling
n It is a nonprobability sampling technique. Items are
included in the sample without known probabilities
of being selected.
n The sample is identified primarily by convenience.
n Advantage: Sample selection and data collection are
relatively easy.
n Disadvantage: It is impossible to determine how
representative of the population the sample is.
n Example: A professor conducting research might use
student volunteers to constitute a sample.

13. 13Slide
Judgment Sampling
n 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.
n It is a nonprobability sampling technique.
n Advantage: It is a relatively easy way of selecting a
sample.
n Disadvantage: The quality of the sample results
depends on the judgment of the person selecting the
sample.
n Example: A reporter might sample three or four
senators, judging them as reflecting the general
opinion of the senate.

14. 14Slide
Snowball Sampling:
Definition
Snowball sampling or chain-referral sampling is defined as
a non-probability sampling technique in which the samples
have traits that are rare to find. This is a sampling
technique, in which existing subjects provide referrals to
recruit samples required for a research study.

15. 15Slide
Types of Snowball Sampling
1.Linear Snowball Sampling: The formation of a
sample group starts with one individual subject providing
information about just one other subject and then the chain
continues with only one referral from one subject. This pattern
is continued until enough number of subjects are available for
the sample.
2.Exponential Non-Discriminative Snowball Sampling: In
this type, the first subject is recruited and then he/she
provides multiple referrals. Each new referral then provides
with more data for referral and so on, until there is enough
number of subjects for the sample.
3.Exponential Discriminative Snowball Sampling: In this
technique, each subject gives multiple referrals, however, only
one subject is recruited from each referral. The choice of a new
subject depends on the nature of the research study.

16. 16Slide
Point Estimation
n 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.
n We refer to as the point estimator of the population
mean .
n s is the point estimator of the population standard
deviation .
n is the point estimator of the population proportion
p.
x
p

17. 17Slide
Sampling Error
n The absolute difference between an unbiased point
estimate and the corresponding population
parameter is called the sampling error.
n Sampling error is the result of using a subset of the
population (the sample), and not the entire
population to develop estimates.
n The sampling errors are:
for sample mean
| s -  | for sample standard deviation
for sample proportion
|| x
|| pp 

18. 18Slide
Example: St. Andrew’s
St. Andrew’s University receives 900 applications
annually from prospective students. The application
forms contain a variety of information including the
individual’s scholastic aptitude test (SAT) score and
whether or not the individual desires on-campus
housing.

19. 19Slide
Example: St. Andrew’s
The director of admissions would like to know the
following information:
•the average SAT score for the applicants, and
•the proportion of applicants that want to live on
campus.
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
computer-generated random numbers

20. 20Slide
n Taking a Census of the 900 Applicants
•SAT Scores
•Population Mean
•Population Standard Deviation
•Applicants Wanting On-Campus Housing
•Population Proportion
  
 ix
990
900



 
 ix 2
( )
80
900
 p
648
.72
900
Example: St. Andrew’s

21. 21Slide
Example: St. Andrew’s
n Take a Sample of 30Applicants Using a Random
Number Table
Since 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 a random
number table. 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.

22. 22Slide
Example: St. Andrew’s
n 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
436 Number already used
etc. etc.

23. 23Slide
n Sample Data
Random
No. Number Applicant SAT Score On-Campus
1 744 Connie Reyman 1025 Yes
2 436 William Fox 950 Yes
3 865 Fabian Avante 1090 No
4 790 Eric Paxton 1120 Yes
5 835 Winona Wheeler 1015 No
. . . . .
30 685 Kevin Cossack 965 No
Example: St. Andrew’s

24. 24Slide
Example: St. Andrew’s
n Take a Sample of 30 Applicants Using Computer-
Generated Random Numbers
•Excel provides a function for generating random
numbers in its worksheet.
•900 random numbers are generated, one for each
applicant in the population.
•Then we choose the 30 applicants corresponding
to the 30 smallest random numbers as our sample.
•Each of the 900 applicants have the same
probability of being included.

25. 25Slide
Using Excel to Select
a Simple Random Sample
n Formula Worksheet
A B C D
1
Applicant
Number
SAT
Score
On-Campus
Housing
Random
Number
2 1 1008 Yes =RAND()
3 2 1025 No =RAND()
4 3 952 Yes =RAND()
5 4 1090 Yes =RAND()
6 5 1127 Yes =RAND()
7 6 1015 No =RAND()
8 7 965 Yes =RAND()
9 8 1161 No =RAND()
Note: Rows 10-901 are not shown.

26. 26Slide
Using Excel to Select
a Simple Random Sample
n Value Worksheet
A B C D
1
Applicant
Number
SAT
Score
On-Campus
Housing
Random
Number
2 1 1008 Yes 0.41327
3 2 1025 No 0.79514
4 3 952 Yes 0.66237
5 4 1090 Yes 0.00234
6 5 1127 Yes 0.71205
7 6 1015 No 0.18037
8 7 965 Yes 0.71607
9 8 1161 No 0.90512
Note: Rows 10-901 are not shown.

27. 27Slide
Using Excel to Select
a Simple Random Sample
n Value Worksheet (Sorted)
A B C D
1
Applicant
Number
SAT
Score
On-Campus
Housing
Random
Number
2 12 1107 No 0.00027
3 773 1043 Yes 0.00192
4 408 991 Yes 0.00303
5 58 1008 No 0.00481
6 116 1127 Yes 0.00538
7 185 982 Yes 0.00583
8 510 1163 Yes 0.00649
9 394 1008 No 0.00667
Note: Rows 10-901 are not shown.

28. 28Slide
n Point Estimates
• as Point Estimator of 
•s as Point Estimator of 
• as Point Estimator of p
n Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
x
p
  
 ix
x
29,910
997
30 30

  
 ix x
s
2
( ) 163,996
75.2
29 29
 p 20 30 .68
Example: St. Andrew’s

29. 29Slide
Sampling Distribution of
n Process of Statistical Inference
Population
with mean
 = ?
A simple random sample
of n elements is selected
from the population.
x
The sample data
provide a value for
the sample mean .x
The value of is used to
make inferences about
the value of .
x

30. 30Slide
n The sampling distribution of is the probability
distribution of all possible values of the sample
mean .
n Expected Value of
E( ) = 
where:
 = the population mean
Sampling Distribution of x
x
x
x
x

31. 31Slide
n 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.
x


x
n
N n
N



( )
1


x
n

( ) / ( )N n N 1
x
Sampling Distribution of x

32. 32Slide
n 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 probability distribution.
n 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
probability distribution.
x
x
Sampling Distribution ofx

33. 33Slide
n Sampling Distribution of for the SAT Scoresx
Example: St. Andrew’s

   x
n
80
14.6
30
E x( )   990
x

34. 34Slide
n Sampling Distribution of for the 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 plus or
minus 10 of the actual population mean  ?
In other words, what is the probability that
will be between 980 and 1000?
x
Example: St. Andrew’s
x

35. 35Slide
n Sampling Distribution of for the SAT Scores
Using the standard normal probability table with
z = 10/14.6= .68, we have area = (.2518)(2) = .5036
x
Sampling
distribution
of x
1000980 990
Area = .2518Area = .2518
Example: St. Andrew’s
x

36. 36Slide
n The sampling distribution of is the probability
distribution of all possible values of the sample
proportion
n Expected Value of
where:
p = the population proportion
Sampling Distribution of p
p
p
p
E p p( ) 

37. 37Slide
Sampling Distribution ofp
p
p
p p
n
N n
N

 

( )1
1
p
p p
n

( )1
p
n Standard Deviation of
Finite Population Infinite Population
• is referred to as the standard error of the
proportion.

38. 38Slide
n Sampling Distribution of for In-State Residents
The normal probability distribution is an acceptable
approximation since np = 30(.72) = 21.6 > 5 and
n(1 - p) = 30(.28) = 8.4 > 5.
p


 p
.72(1 .72)
.082
30
Example: St. Andrew’s
( ) .72E p 

39. 39Slide
n Sampling Distribution of for In-State Residents
What is the probability that a simple random
sample of 30 applicants will provide an estimate of
the population proportion of applicants desiring on-
campus housing that is within plus or minus .05 of
the actual population proportion?
In other words, what is the probability that
will be between .67 and .77?
p
Example: St. Andrew’s
p

40. 40Slide
n Sampling Distribution of for In-State Residents
For z = .05/.082 = .61, the area = (.2291)(2) = .4582.
The probability is .4582 that the sample proportion
will be within +/-.05 of the actual population
proportion.
Sampling
distribution
of
0.770.67 0.72
Area = .2291Area = .2291
p
p
Example: St. Andrew’s
p

41. 41Slide
Properties of Point Estimators
n 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.
•Unbiasedness
•Efficiency
•Consistency

42. 42Slide
Properties of Point Estimators
n Unbiasedness
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.

43. 43Slide
Properties of Point Estimators
n 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.

44. 44Slide
Properties of Point Estimators
n 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.

45. 45Slide
End of Chapter 7