1. St. Edward’s University
Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University

2. Chapter 7, Part A Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of

3. 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

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

12. 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.

13. 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.

14. 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

15. 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.

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

17. 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.

18. 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.

19. 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.)

20. 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

21. 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

22. 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.

23. 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.

24. 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

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. 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.

29. Sampling Distribution of for SAT Scores

30. 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?

31. 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

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

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

34. 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

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

36. 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

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

38. 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:

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

40. 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.

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

42. End of Chapter 7, Part A