Section 1.3
Random Sampling
Population: the collection of all individuals or items under
consideration in a statistical study. The size of the population is
represented by N.
Sample: the part of the population from which information is
obtained. The size of the sample is represented by n. The sample
size is determined more by the number of samples than a
percentage of the population.
What is important in sampling is that the sample is representative
of the population.
How do we do make sure that the sample is representative of the
population?
We can use a simple random sample
Simple random sample: every member of the population is
equally likely to be chosen. Also the members chosen are
independent of each other (not related)
Example 1
If one person is chosen for an experiment, then there siblings are
not automatically chosen in a simple random sample (srs). In
another type of test (test for dependent data) siblings may be
chosen, but not for a simple random sample.
How can we obtain a simple random sample?
Use a random number table or random number generator (on
calculator) Random Number Table is a table of randomly chosen
digits
Procedure 1: Random Number Table Vertical Prodedure
How to choose a random sample using a random number table
1. Number the objects or individuals in the population from 1 to
N where N is the total number of objects in the population.
2. If N is a two digit number, then two digits will be used, If N is
a three digit number, then three digits will be used.
3. The sample size is n.
4. Close your eyes and hit the chart.
5. Go down (or across) the table and find n one, two, or three
digit numbers (depending upon N) between 1 to N
6. Don’t repeat the numbers if a number is chosen twice keep
going until the sample size of n is met.
Example 2
From a population size of 84, a sample size of ten is to be
chosen. Use the random number table to do this.
Procedure 2: Random Number Table Horizontal Prodedure
1. A row from the random number table is chosen.
2. List all of the numbers in that row which is a string of random
digits.
3. If the population size is two digits such as 56, then while
scanning the string of random digits, list all of the numbers in
the string that are between 01 and 56.
4. Numbers greater than 56 such as 57, 75, 83, and 99 would
not be included.
5. Continue along the string selecting numbers until the sample
size is met.
Example 3
The sample size is five and the population is numbered 01 to 56
Select five numbers from the chosen string of
13962709926517278053021908363466012
There are also some nonsimple random sampling techniques that
are used
Random cluster sample: used to generate a sample when
members of a population are geographically far apart from each
other. The area is divided into clusters and random clusters are
chosen. All subjects and objects in the cluster are sampled.
Procedure 3: Cluster Sampling
1. Divide the population into groups (clusters)
2. Number the clusters.
3. Obtain a simple random sample of the clusters.
4. Use all of members of the clusters obtained in Step 3 as the
sample.
Stratified sampling with proportional allocation: The
population is divided into strata. Each of these strata are
considered to be homogenous collections of subjects or objects.
Then a random sample within each strata is taken according to
the number of subjects or objects within the strata. The objects or
subjects from each strata are combined to form the sample for the
study.
Procedure 4: Stratified Sampling with Proportional Allocation
1. Divide the population into subpopulations (strata)
2. Determine how many items/subjects will be included in the
sample from each strata.
3. To determine how many from each stratum are sampled the
following is done. The sample size from each stratum is
calculated by multiplying the total sample size and by the
proportion of items/subjects to the total population size.
4. Each stratum will be numbered off: one through the total
strata size.
5. Randomly select the members from each strata that will be
used to make up the sample.
Example 4: Sampling Dorm Residents
Students in the dormitories of a university live in clusters of four
double rooms, called suites. There are 48 suites with 8 students
per suite.
a. Describe a cluster sampling procedure for obtaining a
sample of 24 dormitory residents.
b. Students typically chose their friends from their classes as
suite-mates. With that in mind, do you think that cluster
sampling is a good procedure for obtaining a representative
sample of dormitory residents? Explain your answer.
c. The university housing office has separate list of dormitory
residents by class level. The number of dormitory residents
each class level is as follows:
Class Level
Freshman
Sophomore
Junior
Senior
Number of Dormitory
Residents
128
112
96
48
Use a table to design a procedure for obtaining a stratified sample
of 24 dormitory residents. Use stratified random sampling with
proportional allocation.
Stratified sampling with proportional allocation is more reliable
than cluster sampling. Ideally, the members of each stratum
should be homogenous relative to the characteristic under
consideration.
With all of our sampling techniques we will not have the exact
same results as if we were using the whole population.
Why aren’t the results the same?
Sampling error
Sampling error or chance error due to sampling: is the
discrepancy between the sample and the population.
Sampling bias: a tendency to more likely chose some items from
the population instead of the others
Example 5
Random number table (always start around the center).d
Historically, only men were used in medical studies.
Non sampling errors types
1. Non sampling errors: an error not caused by the sampling
method. Could be caused by the wording of a question.
2. Non response bias: a person not answering a question in a
survey or not returning a written survey.
3. Missing data: the experimenter planned on having data
from a particular group, but the observations are missing.