h2 sampling methods (indpt module)

7/30/2019 H2 Sampling Methods (Indpt Module)

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Innova Junior College / H2 Mathematics / Sampling Methods JC 1/ 2012

Sampling Methods 1 of 16

Sampling Methods (Independent Module)

Pri nt References

1. Introductory Statistics, Neil A. Weiss2. Introducing Statistics, Graham Upton, Ian Cook3. AP Statistics 2008- 2009, Duane C. Hinders4. A Concise Course in Advanced Level Statistics with Worked Examples,

J. Crawshaw and J. Chambers

5. Elementary Statistics: A Step by Step Approach, Allan G. BlumanNon-print References

1. Sampling Methodshttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htm

2. The following link contains some interactive Excel files that allow you toexperiment and compare the various sampling methods covered in this chapter.

http://www.coventry.ac.uk/ec//~nhunt/meths/ss.html

Objective: At the end of the chapter, students should be able to

1. Understand the concepts of population, sample and random sample;2. Understand the purpose and appropriate use of the 4 sampling methods ( random,

stratified, systematic and quota samples) in practical situations;

3. Explain in simple terms why a given sampling method may be unsatisfactory, andsuggest possible improvements.

1 Introduction

1.1 The Purpose of Sampling

In many practical statistical investigations,

we are interested in gathering information

about a population based on data derived

from a sample of the population.

This enables us to draw conclusions about a

large set of data (population) by studying arelativelysmallamount of data (sample).

For the sample to be useful, it has to be a

representative sample, that is, it has the

essential characteristics of the population

being studied, and is free of any type of

systematic bias.

A sample should be large enough to give a

good representation of the population but

small enough to be manageable.

population (see 1.2.1)

What is a good sample size?

large enough for a good

representation of population

small enough to be

manageable (time & costs

constraints

sample

(see 1.2.2)
http://www.statcan.ca/english/edu/power/ch13/probability/probability.htmhttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htmhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htm


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1.2 Definitions

1.2.1 Population

is a complete collection of

individuals or items about which

statistical information is desired.1.2.2 Sample

is a finite subset of a population.

1.2.3 Sampli ng Frame

is a list of all members of the

population from which we can draw

a sample.

1.2.4 Random Sample

is a sample drawn from a population

such that every member of thepopulation has an equal chance of

being selected.

Subset means part of/contains some

members of.

2 Sampling Methods

In this chapter, we shall study

the following 4 sampling

methods:

(simple) random sampling

systematic sampling stratified sampling

quota sampling

The first 3 methods will produce

a random sample, whereas

quota sampling will produce a

non-random sample.

population

What methods can used to collect a sample?

(simple) random sampling quota sampling

systematic sampling

stratified sampling


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2.1 (Simple) Random Sampling

In (simple) random sampling, every

possible sample of the same size has an

equal chance of being selected.

Also, each member of the population has an

equal chance of being included in the

sample.

2.1.1 Method for (Simple) Random Sampling

Suppose we want to obtain a sample of size

n from a population of sizeN.

1. Number the list of all the members of

the population from 1 toN.

2. Make a random selection ofn of these

numbers.

(This can be done by generatingrandom numbers with a computer or

GC using the command

randInt(1, N, n).)

3. The members corresponding to these

n numbers will constitute the sample.

Example of a contextual situation where the

simple random sampling method can be

carried out:

A total of 100 students from IJC attended

an enrichment class at Science Centre, fromwhich a simple random sample of 5

students are selected to answer a

questionnaire on the effectiveness of the

programme.

1. All the 100 students could each be

assigned a number based on

alphabetical order of their first names.

2. Generate 5 random integers between 1

and 100 using the GC command

randInt(1, 100, 5).

3. The students corresponding to the

randomly selected numbers will form

the sample.

Example:

12

34

5

9899

100

population of size, N =100

population of size, N =100

1. Ang Chee Yong

2. Chan Chee Tat

3. Cheong Chong Yan

4. ..

..99. Yang Yan Yan

100. Zaina Bte Aman

1

Use computer or GC to generate

n= 5 numbers at random2

21172

101

32138

101

46316

1764

32338

11100

Some of the many samples of size n= 5.

Each of these samples has the same

probability of being selected.

A simple random

sample of size, n =5

4. Cheong Chong Yan

25. Feng Tian

78. Lee Mary

89. Tan Mei Mei

99. Yang Yan Yan

3


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2.1.2 Use of the GC to generate random integers

The syntax for generating n random integers between m andNinclusive is as

follows:

randInt(m,N, n)

Example:

Generate 5 random integers between 1 and 100 inclusive

Keystrokes/Steps Screen Display

Step 1:

Start with the home screen.

Press .

Step 2:

Scroll right to PRBby pressing .

Press to select the function

5:randInt(.

Step 3:

Key in 1 for the lowest value of the

integers to be randomly generated, 100 for

the highestvalue, and 5 for the number of

integers to be randomly generated.

Press to generate the 5 random

integers.

Hence the 5 random integers generated are95, 91, 15, 52 and 41.

Lowest value of the

integers to be

randomly generated.

Highest value of the

integers to be

randomly generated.

Numberof integers

to be randomly

generated.


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2.1.3 Advantages and Disadvantages of Simple Random Sampling

Advantages Disadvantages

Method can be carried out

quickly and easily with the help

of a computer or GC to generaterandom numbers.

Sample obtained may happen to

be one that is not representative

of the population (see example1(iii) for illustration).

Suitable only if the sampling

frame is complete and up-to-

date.

Example 1:

In a large country ASU, an opinion poll which concerned the support for the

two presidential candidates Macane and Obaba was carried out before the

presidential election. A simple random sample of 1000 people was interviewed

and based on the results of the poll, an easy win was predicted for Macane.

(i) Describe how a simple random sample of 1000 can be selected.

(ii) Give one disadvantage of simple random sampling in the context of the

question.

(iii) When the actual election result was out, it turned out that Obaba won the

election. Give one possible reason for this difference from the predicted

result.

Solution

(i) Obtain the sampling frame of all

eligible voters from the Electoral

Board.Assign a number to each person in

the sampling frame. Generate 1000

random numbers with a computer.

The people corresponding to these

1000 numbers will constitute a

simple random sample of 1000.

Need to mention what the sampling frame

(all eligible voters in country ASU) is.

Need to assign a number to each member in

the sampling frame before generating n

(sample size) random (must mention this

word) numbers from the list of assigned

numbers.

Need to mention that the people that have

been assigned the chosen numbers make up

the selected sample.

(ii) Possible disadvantages:

As the country is large, it maybe expensive to travel to the

different parts of the country to

interview those sampled.

Some of those selected may not

be available for interview.

You may write any one of the possible

disadvantage BUT must ensure that you

write in context, and not the theoretical

answer found in 2.1.3.

(iii) Since the population is very large, a

sample size of 1000 may not result in

a representative sample. In fact, the

sample chosen could happen to have

been made up of a greater proportionof Macanes supporters.

This is an important disadvantage of simple

random sampling.


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2.2 Systematic Sampling

Systematic sampling is carried out by systematically drawing the sample from

the population.

2.2.1 Method for Systematic SamplingSuppose we want to obtain a sample of size n from a population of sizeN.

1. Listall members of the population in some order (usually with reference to

context and/or purpose of study).

For example, we can have

the pupils in a school arranged according to the school register if we want

to choose a sample to interview them for their opinion on the school

canteen food,

the order in which customers enter a particular store if we want to survey

how long they spend in the shop.

2. Determine the sampling in tervalN

kn

(Round down to the nearest integer if

necessary)

3. Choose the first member of the sample at random from the first kmembers,

i.e., choose an integer randomly from the numbers 1 to k.

4. Choose every other kth

member after the first member is chosen.

Illustration with an example:

Use systematic sampling to choose a sample of size 30 from the 400 pupils in a school

to interview them for their opinion on the school canteen food.

List all the students according to the school register.

Sampling interval, k =400

13.3 1330

Suppose the random number selected in is 2.

131 2 3 400..391

1

14 15 16 26

2

272829 39

3

378379380 390

30..

1

2

4 3

3Choose a

number

from 1 to 13at random.

131 2 3 400..391

1 3

378379380 390

30

14 15 16 26

2

272829 39

..

215 28

379

..

A systematicsample of

size 30

13 13 13


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2.2.2 Advantages and Disadvantages of Systematic Sampling


Sample is more evenly spread

over the population.

Sample may be biased when the

order in which members of the

population are lined up have acyclic pattern.

Example 2:

A student councilor in a junior college is to conduct a survey of the graduating

class concerning plans for Graduation Night. There are 800 students in the

graduating class and he needs a sample of 80.

(i) Describe how a systematic sample may be taken.

(ii) Explain why the sample selected in (i) is a random sample.

Solution

(i) Get a list of all students from the

graduating class arranged according

to the school register (by classes).

Sampling interval k

Choose the first student at random

from the first 10 names.

Choose every 10th

name after the first

student has been chosen until a sample

of 80 is obtained.

It is meaningful to list the students by

classes in the school register order as

the information needed concerns the

plans for Grad Night, and generally

students from a more Science based

subject combination may have different

preference from that of a more

Humanities based subject combination.

With systematic sampling performed on

this list, the sample will be one that ismore representative of the whole

graduating batch of students.

(ii) Since the first member is selected at

random from the first 10 names, the

first 10 students in the register have a

probability of1

10of being selected.

The subsequent members of the

sample are fixed once the first member

has been selected. So every member inthe population has an equal chance,

which is1

10, of being selected to be in

the sample. Thus a systematic sample

is a random sample

Recall that

simple random sampling

systematic sampling

stratified sampling (section 2.3)

produce random samples.


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Example 3:

A company has 20 departments, each of which has 20 employees. A list of

employees is ordered by department, with the manager listed first, and then the

other employees in descending order of seniority.

A survey is conducted to find out about the employees opinions on their workenvironment. A systematic sampling approach was used to obtain a sample of

size 20. Give one disadvantage of such an approach in this case.

Solution

As the sample size is 20,

sampling interval is

k .

Thus every department

will have an employee

being selected.However, due to the

cyclic pattern in which

the employees are listed,

it will result in a sample

consisting of employees

of the same seniority(e.g. only managers or

only junior employees

are chosen).

The sample will thennot be representativeof

all the employees

opinions on their work

environment.

Suppose 19is the number obtained from the random selection

of numbers from 1 to 20 (first twenty numbers)

Note that you must give the disadvantage in context (see words

in bold in solution) and not merelythe technical answer cyclic

pattern as stated in the 2.2.2.

20

20

M AM supv grp L JJ

Department 1

M AM supv grp L JJ

Department 2

M AM supv grp L JJ

Department 3

M AM supv grp L JJ

De artment 19

M AM supv grp L JJ

De artment 20


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2.3 Stratified Sampling

Many populations contain identifiable

strata, which are distinctive non-

overlapping subsets of the

population.

Strata might be based on gender, age

groups, ethnic groups etc.

Stratified sampling ensures that the

proportions of the population falling

into these strata are reproduced by

the sample. Thus a stratified sample is

likely to be a representative sample of

its population.

However, a simple random sample is

unlikely to reproduce these proportions

and thus may provide a biased and

non-representative view of the

population.

Gender, age groups, ethnic groups can be strata

as a person can only be in 1 strata. For example,

a person is either a male or female and not

both!

You may view a stratified sample as a

miniature version of the population as the

proportions are the same.

A sample obtained from simple random sampling

may happen to be one with a greater

proportion of females which is not a good

representation of the population.

2.3.1 Method for Stratified Sampling

1. Divide the population intomeaningful strata.

Note: This method requires the

sampling frame, together with

additional information on the

chosen strata, which will give the

proportion for each stratum.

2. Draw random samples separately

from each stratum, with samplesize proportional to the relativesize of the stratum.

3. These separate random samples are

put together to form the sample of

the population.

The strata must be meaningful to the purposeof the study.

For example, the strata for survey of

preference for type of movies could be gender

as boys and girls generally prefer different

genre.

sample

male female

populationmale female

Sample (size = 12)

S1 S2 S3

435

population (size = 120 )strata

2 (S2)

strata 1

(S1)

50 30

strata 3

S3

40


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2.3.2 Advantages and Disadvantages of Stratified Sampling


Likely to give a good

representative sample (for the

purpose of study) of thepopulation.

Strata may be difficult to define

clearly.

Example 4:

In a company, the proportions of staff in different age-groups are as follows:

Age under 40 between 40 and 60 over 60

Percentage 38 40 22

Describe how a stratified sample of 20 staff can be obtained.

Solution

Get a list ofall staff members. Divide them

into age-groups under 40, between 40 and

60, over 60.

Age under 40between

40 and 60over 60

Number

of staff

3820

100

7.6 8

40

20 8100

22

20100

4.4 4

From each age group, draw a random

sample of the required size as shown in the

table.

The three random samples drawn will

together form a stratified sample of 20

staff.

For stratified sampling, additional info

on the chosen strata (age in this case) is

required.

You must show how the sample size in

each strata is obtained.

Note that the numbers must be rounded

to the nearest integer because number

of people must be whole numbers. Also,

the total number must be the requiredsample size.

It is important that simple random

sampling is conducted within each

stratum.

Need to say that the random samples

from the strata together form the

required sample which is a miniature

version of the population with the sameproportions.


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2.4 Quota Sampling

Quota sampling is similar to

stratified sampling in the

sense that the population is

also divided into distinctive,non-overlapping strata.

However, the selection of

respondents within the strata

is left to the interviewer, and

thus is non-random.

Stratified sampling Quota sampling

Divide population into non-overlappingstrata (which is

meaningful to the purpose of study)Requiressampling frame

and information on the

strata chosen.

Does not require a sampling

frame.

The sample size for each

stratum is proportional

to the relative size of

the stratum.

A quota for each stratum is

specified (without

reference to population) to

define the sample.

Random sample selected

within each stratum.

Selection of sample within

each stratumis

non-random, dependent oninterviewer/convenience.

2.4.1 Method for Quota Sampling

1. Divide the population into

meaningful strata.

Note: For Quota Sampling, unlike

stratified sampling, a sampling

frame is not required.

2. A quota for each stratum is

specified to define the sample.

3. The interviewer will select a

sample that meets the

requirements of the quota.

Like in stratified sampling, the strata must be

meaningful to the purpose of the study.

The quota is pre-determined.

The interviewer can choose any person, at his

discretion/convenience, that meets the criteria

stipulated by the quota.


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Example of a contextual situation where the

simple random sampling method can be carried

out:

In a market survey, an interviewer is given an

assignment to interview 20 people in the streetbased on the following quota.

Sex Age Social Class

Male 9 20 29 4 High 2Female 11 30 44 6 Middle 4

45 64 7 Low 1465+ 3

The choice of the sample is subjective. A possible

choice that meets the quota set out is as follows:

Social Class

High Middle Low

Sex

/AgeM F M F M F

20 29 1 1 1 1

30 44 1 1 3 1

45 64 1 1 2 3

65+ 1 2

This is the quota requirements set

out for the sample.

This is a possible choice because

No. of males = 1+1+1+3+2+1=9

No. of females

= 1+1+1+1+1+1+3+2=11

No. with age 20-29 =1+1+1+1=4

No. with age 30-44 =1+1+3+1=6

No. with age 45-64 =1+1+2+3=7

No. with age 65+ =1+2=3

No. of high social class =1+1=2

No. of middle social class

=1+1+1+1=4

No. of low social class=1+3+2+1+1+1+3+2=14

Note that the interviewer can have

other choices but must ensure that

the quota requirements are met.

2.4.2 Advantages and Disadvantages of Quota Sampling


Does not require a sampling

frame.

Information can be collectedquickly and easily.

Lower cost of implementation as

it is up to interviewers

convenience.

Sample obtained not random.

Sample is not a good representative of

the population as compared with othertypes of sampling.

Bias as the interviewer may simply

select those who are easiest to

interview.

However, quota sampling is widely used

in market research, street interviews and

opinion polls. It is very useful and

convenient though it is non-random in

nature.

Note:Information gathered from quotasampling should be treated with caution.


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Example 5:

A market researcher has been commissioned to find out how many times per

month members of the public visit their heartland shopping malls. She intends

to take a quota sample of 80 adults in the 18 to 65 age range.

Suggest suitable strata and describe how a quota sample may be taken. Give oneadvantage and one disadvantage of quota sampling.

Solution

As shopping habits depend on the age and

gender, the strata to be considered are age group

and gender.

Divide the population into age groups of

1825, 2635, 3655, 5665.

Within each age group, take equal number of

males and females.A possible quota for each stratum is as follows:

Age group

18

25

26

35

36

55

56

65Total

Male 10 10 10 10 40

Female 10 10 10 10 40

Total 20 20 20 20 80

The market researcher selects a sample such that

the quota is met.

Possible advantages

Quota sampling does not require a sampling

frame. Therefore quota sampling is

appropriate in this case because it would be

difficult/impossible to obtain a sampling

frame of all the shoppers visiting the mall.

Quota sampling is easy to administer and the

information can be collected quickly.

Quota sampling involves lower cost as

compared to other types of sampling methods

because it only requires minimal logistical

preparation.

Possible disadvantages

The sample is biased as the interviewer may

select those who are easiest to interview.

The sample is likely not a good representative

of the population.

The strata are chosen meaningfully

so that it addresses the purpose of

the survey.

When describing the method of

conducting quota sampling, need to

state the

strata & categories quota: the number to be chosen

for each stratum

State that the market researcher

selects a sample that meets the

requirements of the quota set out.


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Example 6:

There are 29 students in class A, 32 students in class B, 34 students in class C

and 25 students in classD. The teacher wants to choose a sample of 20 students

from these 4 classes. He decides to select 5 students at random from each class.

(i) Explain why the sample formed is not a random sample.(ii) Suggest and describe a method to obtain a random sample of 20 students

from all the 4 classes.

Solution

(i) The sample obtained is not a random sample because

each student from the 4 classes does not stand an equal

chance of being selected to be in the sample.

The probability of choosing a student from class A, B,

CandD are5

29

,5

32

,5

34

and5

25

respectively.

The probability of choosing

5 students at random from

a class of 29 students is

5

29.

(ii) The teacher may want to use stratified sampling to

obtain the required random sample.

Get a list of all the 120 students from the 4 classes.

Categorize them according to theirclasses.Class A B C D

No. of

students

2920

120

4.83

5

3220

120

5.33

5

3420

120

5.67

6

2520

120

4.17

4

From each class, draw a random sample of the

required size as shown in the table.

The four samples drawn will together form a

stratified sample of 20 students from the 4 classes.

Recall that a stratified

sample is a random sample.

The strata used here is

the class.

Summary

Sample

frame

Random Advantage Disadvantage

Random

Sampling

Yes Yes Quick and easy

to carry out

Difficult to get

sample frame

Systematic

Sampling

No Yes More evenly

spread

Maybe Cyclic, not

representative

Stratified

Sampling

Yes Yes Good

representative

Difficult to define

and obtain strata

Quota

Sampling

No No Easy and

economical

Non-random,

maybe biased


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Sampling Methods (Tutorial)

1 (2010/Prelim/MI/10)

A group of 10 students is to be selected for an overseas exchange programme

from an institute of 3 pre-university (PU) levels, with a total enrolment of 1000

students.

(i) Describe how the selection may be done using simple random sampling

and suggest an advantage of this sampling method in the context of the

question. [3]

(ii) Write down one disadvantage of this method of selection, citing an

example in the context of the question. [2]

(iii) State and describe a sampling method that gives a better representation

of each pre-university level in the selection process, given that there are

500 PU1 students, 300 PU2 students and 200 PU3 students.

[3]

2 (H2 Maths/9740/N08/II/5)

A school has 950 pupils. A sample of 50 pupils is to be chosen to take part in a

survey. Describe how the sample could be chosen using systematic sampling.

[2]

The purpose of the survey is to investigate pupils opinions about the spo rts

facilities available at the school. Give a reason why a stratified sample might be

preferable in this context. [2]

3 (H2 Maths/9740/N09/II/5)

A cinema manager wishes to take a survey of opinions of cinema-goers.

Describe how a quota sample of size 100 might be obtained, and state one

disadvantage of quota sampling. [3]

4 (H1 Maths/8863/N09/I/11(a))

An insurance company receives a large number of claims for flood damage. On

a particular day the company receives 72 such claims. Because of staff

shortages, it is only possible to process 8 of these claims.

(i) Describe how you would choose a systematic random sample of size 8

from the received claims. [2]

(ii) Comment on whether this method of sampling gives a better indication

of the value of the 72 claims as compared to simply choosing as the

sample the first 8 claims received. [1]


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5 An employment agency wants to estimate the number of unemployed people in

an HDB new town, and so decide to obtain information from a sample of its

adults who are of working age.

(i) Give one reason why it would not be appropriate to obtain the sample by

stopping adults at the town centres supermarket during one workingday.

(ii) The new town has 10 000 households and they can be classified as

follows.

Housing type No. of household

3-room 1250

4-room 4500

5-room 2000

Executive 1750

Others 500

Explain, using suitable calculations, how the agency could use this

information to obtain a representative sample of 1000 households.

(iii) In an attempt to obtain a representative sample of people, the agency

considers two options:

(A) to interview all the working age adults in the chosen household;

(B) to interview one such person from each household.

The agency decided on option A. Give one reason in favour of this

decision.

6 (H2 Maths/9740/N07/II/5)

(i) Give a real-life example of a situation in which quota sampling could be

used. Explain why quota sampling would be appropriate in this

situation, and describe briefly any disadvantage that quota sampling has.

[4]

(ii) Explain briefly whether it would be possible to use stratified sampling

the situation you have described in part (i). [1]

7 A certain private estate contains many small houses (with small gardens) and a

few large houses (with large gardens). A sample survey of all houses is to be

carried out in this estate. A student suggests that the sample could be selected

by sticking a pin into a map of the estate the requisite number of times, while

blindfolded.

(i) Give two reasons why this method does not produce a random sample.(ii) Describe a better method.

h2 sampling methods (indpt module)

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