Sampling and Sampling Distributions

1. 1Slide
Sampling and Sampling Distributions
n Simple Random Sampling
n Point Estimation
n Introduction to Sampling Distributions
n Sampling Distribution of
n Properties of Point Estimators
x
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
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 .
x

5. 5Slide
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
|| x

6. 6Slide
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.

7. 7Slide
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

8. 8Slide
n Taking a Census of the 900 Applicants
•SAT Scores
•Population Mean
•Population Standard Deviation
•Applicants Wanting On-Campus Housing
  
 ix
990
900



 
 ix 2
( )
80
900
Example: St. Andrew’s

9. 9Slide
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.

10. 10Slide
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.

11. 11Slide
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

12. 12Slide
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.

13. 13Slide
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.

14. 14Slide
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.

15. 15Slide
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.

16. 16Slide
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

17. 17Slide
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

18. 18Slide
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

19. 19Slide
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

20. 20Slide
n Sampling Distribution of for the SAT Scoresx
Example: St. Andrew’s

   x
n
80
14.6
30
E x( )   990
x

21. 21Slide
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

22. 22Slide
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

23. 23Slide
End