1. Chapter 7 Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of
Sampling Methods
n = 100
n = 30

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.
The sample results provide only estimates of the values of the population characteristics.
A parameter is a numerical characteristic of a population.
With proper sampling methods, the sample results will provide “good” estimates of the population characteristics.

3. Simple Random Sampling
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.

4. Simple Random Sampling
Finite Population
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.

5. 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.

6. Simple Random Sampling
Infinite Population
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.

7. 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.

8. Point Estimation
When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased.

9. Sampling Error
The absolute 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 to develop estimates.
The sampling errors are:
for sample mean
for sample standard deviation
for sample proportion

10. Example: St. Andrew’s St. Andrew’s College 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.

11. 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.

12. 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 computer-generated random numbers

13. Example: St. Andrew’s Taking a Census of the 900 Applicants SAT Scores
Population Mean
Population Standard Deviation

14. Example: St. Andrew’s Taking a Census of the 900 Applicants
Applicants Wanting On-Campus Housing
Population Proportion

15. Example: St. Andrew’s Take a Sample of 30 Applicants
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 the textbook’s random number table.

16. Example: St. Andrew’s Take a Sample of 30 Applicants
Using 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.

17. Example: St. Andrew’s Use of Random Numbers for Sampling
3-Digit Applicant
Random Number Included in Sample
No. 744
No. 436
No. 865
No. 790
No. 835
Number exceeds 900
No. 190
436 Number already used
etc etc.

18. Example: St. Andrew’s Sample Data Random
No. Number Applicant SAT Score On-Campus
Connie Reyman Yes
William Fox Yes
Fabian Avante No
Eric Paxton Yes
Winona Wheeler No
Kevin Cossack No

19. Example: St. Andrew’s 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.

20. Using Excel to Select a Simple Random Sample
Formula Worksheet
Note: Rows are not shown.

21. Using Excel to Select a Simple Random Sample
Value Worksheet
Note: Rows are not shown.

22. Using Excel to Select a Simple Random Sample
Value Worksheet (Sorted)
Note: Rows are not shown.

23. Example: St. Andrew’s Point Estimates as Point Estimator of 
as Point Estimator of p

24. Example: St. Andrew’s Point Estimates
Note: Different random numbers would have
identified a different sample which would have resulted in different point estimates.

25. 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 .

26. 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

27. 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.

28. 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 probability 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 probability distribution.

29. Example: St. Andrew’s
Sampling Distribution of for the SAT Scores

30. Example: St. Andrew’s 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?

31. Example: St. Andrew’s Sampling Distribution of for the SAT Scores
1000
980
990
Area = ?

32. Example: St. Andrew’s Sampling Distribution of for the SAT Scores
Using the standard normal probability table with
z = 10/14.6= .68, we have area = (.2518)(2) =
The probability is that the sample mean will be within +/-10 of the actual population mean.

33. Example: St. Andrew’s Sampling Distribution of for the SAT Scores
1000
980
990
Area = 2(.2518) = .5036

34. 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

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

36. Sampling Distribution of
The sampling distribution of can be approximated by a normal probability 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

37. 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 large (approaching 1) values of p, much larger samples are needed.

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

39. Example: St. Andrew’s
Sampling Distribution of for In-State Residents

40. Example: St. Andrew’s 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?

41. Example: St. Andrew’s Sampling Distribution of for In-State Residents
0.77
0.67
0.72
Area = ?

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

43. Example: St. Andrew’s Sampling Distribution of for In-State Residents
0.77
0.67
0.72
Area = 2(.2291) = .4582

44. Sampling Methods Stratified Random Sampling Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling

45. 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. homogeneous group).
A simple random sample is taken from each stratum.
Formulas are available for combining the stratum sample results into one population parameter estimate.

46. 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, etc.

47. 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.
… continued

48. Cluster Sampling
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.
Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.

49. 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.
This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.
… continued

50. Systematic Sampling
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.

51. 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.
Advantage: Sample selection and data collection are relatively easy.
Disadvantage: It is impossible to determine how representative of the population the sample is.
Example: A professor conducting research might use student volunteers to constitute a sample.

52. 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.
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.
Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

53. End of Chapter 7