Statistics and probability

1. SAMPLING PROCEDURE
ABDULRAHMAN M. ISMAEL
Senior High School Department
Philippine Engineering and Agro-industrial Colllege, Inc. Marawi City
Ismael.am67@s.msumain.edu.ph
February 1, 2022

2. II. Sampling Procedures
In sampling, only a relatively small number of respondents or
experimental units will be involved, thus, it is commonly used in practice. We
examine some of the advantages for doing so.
1. It entails lesser cost, lesser effort and it is less time consuming.
a. Since the size of the sample is small compared to the population, the time,
cost and effort involved on a sample study are much less than the study done on
population. For population, huge fund is required because of the resources to be used which
may include more manpower and materials.
b. It will also take a much shorter period of time to gather data from a sample than
from a population. Thus, sampling can lead to more well-timed results as well.
Advantages of Sampling

3. 2. It is less cumbersome and more practical to administer.
It is easier to handle and manage and not as much burden in your
part if you take only data from a smaller number of respondents.
3. Some experiments are destructive so it is not possible to involve
the whole population.
For example, a car manufacturer might want to test the durability of
cars being produced. Obviously, each car could not be crash-tested to
determine its durability or else the company has nothing to sell
anymore. To overcome this problem, samples are taken from
populations, and estimates are made about the total population based on
information derived from the sample.

4. Sampling also has disadvantages, the biggest of which is that the
sample may not truly reflect the characteristic of the population and this
would lead to wrong conclusions. Hence, care must be taken in
choosing a sample. Also, a sample must be large enough to give a good
representation of the population, but small enough to be manageable.
A. Probability Sampling or Random Sampling
In probability sampling, each element has a known probability of
selection, and a chance method such as “draw lots” or using numbers
from a random number table is used in selecting the specific units to be
included in the sample.
Types of Sampling Procedures

5. (1) Simple Random Sampling (SRS)
This is the simplest form of random sampling where every subset
of size n of the population has an equal chance of being selected.
A simple random sample can be done using the “fishbowl”
method, “draw lots” method or using random numbers. In drawing a
simple random sample, the researcher is in effect mixing up the units in
the population before a sample of n units is selected
Steps in Simple Random Sampling (SRS):
(1) Assign a number to each element of the population using the
numbers from 1 to N.
(2) Select n numbers from 1 to N using random process like fishbowl
method or draw lots, or you can use random numbers which can be
generated by a scientific calculator, or you can use table of random
numbers.

6. (a) Each name in a telephone book could be numbered sequentially. If
the sample size is to include 1,000 people, then 1,000 numbers could
be randomly generated by computer or numbers could be picked out
using a random process, for instance, draw lots. These numbers
could then be matched to names in the telephone book, thereby
providing a list of 1,000 people.
(b) Choose a random sample of five (5) students from the following 30
students using the random number of your calculator.
Examples:

7. A disadvantage of simple random sampling is that we can never be assured that all
sectors or groups are represented in the sample. For instance, in Example (b) above,
there is a possibility that all elements drawn will be girls or all will be boys.
To avoid the above mentioned possibility, we need to contemplate and employ other
sampling procedures that can lead to more representative sample in which the
sample units are spread evenly over the entire population. This sampling procedure
is called systematic random sampling.
(2) Systematic Random Sampling
This is also called interval sampling. It means that there is a gap
or interval between each selection. Researchers obtain systematic
samples by numbering each subject of the population and then selecting
every k the element in the population where the first unit is chosen at
random.

8. Steps in taking a Systematic Random Sample:
(1) Assign a number to each element of the population using the
numbers from 1 to N.
(2) Determine the sampling interval k:
NOTE: If k is not a whole number, then it is rounded to the nearest
whole number.
For example, suppose N = 400 and n =15 then k = 400/15 is equal to
26.67. That is, 26.67 is rounded-off to nearest whole number 27.
(3) Select a random start r where 1≤ r ≤ k. The sample will include the r
th element, (r+k)th , (r +2k)th, (r +3k)th and so on until you reach the
desired sample size.

9. (a) If a systematic sample of 6 students were to be selected in a class with an
enrolled population of 48, the sampling interval would be: k = N/n =48 / 6 =
8
All students would be assigned sequential numbers. The starting point
would be chosen by selecting a random number between 1 and 8. If this
number is 5, then the sample will consist of ( rth=5th) element which is the 5
th student, (r+k = 5 +8)th element which is the 13th student, (r +2k = 5
+2(8))th element which is the 21st student, (r +3k = 5 +3(8))th which is the
29th student,(r +4k = 5 +4(8))th which is the 37th student until (r +5k = 5
+5(8))th which is the 45th student.
(b) In a population of 1200 individuals, choose a systematic random sample
of size 9. Solution: k = N/n = 1200/ 9 = 133.33 and since it is not a whole
number, we need to round it off to nearest whole number, that is 133. Since k
=133, we have r = 1, 2, 3,…, or 133. If we choose r =3, the sample points will
be the 3 rd person, the 136th person, the 139th person, the 142nd person, the
145th person, the 148th person, the 151st person, the 154th person until the
157th person.
Examples:

10. NOTE: The list from which the systematic sample is drawn should be
examined that it must not have a periodic pattern because it could
possibly result to a biased sample.
(3) Stratified Random Sampling
In this sampling procedure, the population of N units is first
divided into homogeneous subpopulations called strata (homogeneous
with respect to the characteristics of interest) and then a sample is drawn
from each stratum. This type of sampling assures that all groups or
strata are represented in the sample. Some stratification variables
commonly used by the Social Weather Station (SWS) survey are
location, age and sex. Other stratification may be religion, academic
ability or marital status.

11. Steps in taking a Stratified Random Sample:
(1) Classify the population into at least two homogeneous strata. The basis
for classification must be closely related to the variable of interest.
Suppose we are interested to determine the students’ opinion on the
tuition fee increase, it may be logical to subdivide the population of
students by income of parents, college, or by year level, or by tribe or a
combination of these.
(2) (2) Draw a sample from each stratum by simple or systematic random
sampling.
How many shall we take from each stratum? The most commonly used
formula is proportional allocation. In proportional allocation, the number of
units to be taken from each stratum is proportional to the size of the
subpopulation; that is, between two strata of different sizes, a bigger sample
will be taken from the bigger stratum.
Proportional Allocation. If the size N of the population is divided into k
homogeneous subpopulations or strata of sizes N1, N2, …, Nk, then the
sample size to be taken from each stratum i is obtained using the formula

12. NOTE: If n_i is not a whole number, then it is rounded-off to the nearest
whole number.
(a) The manager of a girls’ dormitory wants to learn how the students feel
about the dorm’s services. The students were classified according to the
following scheme:
If we use proportional allocation to select stratified random sample of size n
= 40, how large a sample must be taken from each stratum?
Examples:

13. (b) In an election survey in Makati City, registered voters are classified
according to the following scheme:
If one uses proportional allocation to select a stratified random sample of
size n=345, how large a sample must be taken from each stratum?
Solution;

14. Solution;
(4) Cluster Sampling
Cluster sampling assumes that the population is naturally separated by groups or
clusters. A number of clusters are selected randomly and then all or parts of the
units within the selected clusters are included in the sample. No units from the
non-selected clusters are included in the sample. It differs from stratified
sampling, because in the latter, sample units are selected from every group.
You may be able to save much resource in cluster sampling compared to
SRS or Stratified Random Sampling, but cluster sampling leads to less precise
estimates. This is because when we sample each unit in a cluster, we would expect
to get similar information which may be different from other clusters not selected.

15. Steps in taking Cluster Sampling:
(1) Divide the population area into clusters.
(2) Select randomly a few of these clusters.
(3) Choose all the elements from the clusters selected or select only a portion
of it.
(a) Suppose the population of a study is residents of a condominium in a
large city. If there are 10 condominium buildings in this city, the
researcher can select two buildings randomly from the 10 and interview
all (or a subsample) of the residents from these buildings.
(b) Suppose an organization wishes to find out which sports senior students
are participating in the Philippines. It would be too costly and would take too
long to survey every student, or even some students from every school.
Instead, 100 schools are randomly selected from all over the Philippines.
These schools are considered to be clusters. Then every senior student in
these 100 schools is surveyed. In effect, students in the sample of 100 schools
represent all Senior students in the Philippines.

16. The advantages of Cluster sampling are: reduced costs; simplification
of the fieldwork and more convenient administration. Instead of having
a sample scattered over the entire coverage area, the sample is more
localized in relatively few centers. However, it often gives less accurate
results due to higher sampling error than for simple random sampling
with the same sample size. In the above example, you might expect to
get more accurate estimates from randomly selecting students across all
schools than from randomly selecting 100 schools and taking every
student in those chosen schools.

17. THE END