Sampling and Sampling Distributions

Sampling and Sampling
Distributions
Part2
Recall….
• We introduced two methods for collecting data;
Experiments Vs. Surveys.
• In an experimental study, a variable of interest is first

identified. Then one or more other variables are
identified and controlled so that data can be obtained
about how they influence the variable of interest.
• Surveys may be administered in a variety of ways, e.g.

Personal Interview, Telephone Interview, Self
Administered Questionnaire.
Recall….
• Target population: the population about which we want
to draw inferences.
• A frame A list, map, directory, or other source used to

represent the population.
• Statistical inference permits us to draw conclusions

about a population parameter based on a sample.
• Now, the question is: how a sample will be taken from

a population? What are the different techniques for
selecting a sample? What is the difference between
a random and a non-random sample??
Lecture Outline
• Random Vs. Non Random Sampling.
• Random Sampling Techniques.
• Sampling and Non-Sampling Errors.
• Exercises.
Random Vs. Non Random
Sampling.
Random sampling (Probability Sampling):
• Random sampling implies that chance enters into the

process of selection.
Nonrandom sampling (Non-probability sampling ):
• Members of nonrandom samples are not selected by

chance. For example, they might be selected
because they are at the right place at the right time or
because they know the people conducting the
research.
Sampling.
Example:
• A non-random sample of homes for door-to-door

interviews might include houses where people are at
home, houses near the street, houses with no dogs,
first-floor apartments, and houses with friendly people.
• In contrast, a random sample would require the

researcher to gather data only from houses and
apartments that have been selected randomly, no
matter how inconvenient or unfriendly the location
was.
Sampling.
• Nonrandom sampling methods are not appropriate

techniques for gathering data to be analyzed by most
of the statistical methods presented in this text.
• The statistical methods presented and discussed in

this text are based on the assumption that the data
come from random samples.
Random Sampling Techniques
• We will focus our attention on three basic

random sampling techniques:
1. Simple Random Sampling.
2. Stratified Random Sampling.
3. Systematic Random Sampling.

1.Simple Random Sampling…
• Number each frame unit from 1 to N.
• Use a random number table or a random number generator

to select n distinct numbers between 1 and N, inclusively.
• Easier to perform for small populations and difficult to apply

for large populations (The process of numbering all the
members of the population and selecting items will be
difficult).
• E. g : Assign a number to each student in the class, write

these numbers on identical pieces of paper, place them in a
hat and draw 3 pieces of paper from the hat. Any group of
three students is equally likely to be chosen.
• Example:
• Use simple random sampling to select a
sample of six companies.
• First, we number every member of the
population as follows:
01 Alaska Airlines 11 DuPont 21 Lucent

02 Alcoa 12 Exxon Mobil 22 Mattel
03 Ashland 13 General Dynamics 23 Mead
04 Bank of America 14 General Electric 24 Microsoft
05 BellSouth 15 General Mills 25 Occidental Petroleum
06 Chevron 16 Halliburton 26 JCPenney
07 Citigroup 17 IBM 27 Procter & Gamble
08 Clorox 18 Kellog 28 Ryder
09 Delta Air Lines 19 KMart 29 Sears
10 Disney 20 Lowe’s 30 Time Warner
• Six different two-digit numbers must be
selected from the table of random numbers.
• N=30, n=6
Because this population contains only 30
companies, all numbers greater than 30 (31–99)
must be ignored
9 9 4 3 7 8 7 9 6 1 4 5 7 3 7 3 7 5 5 2 9 7 9 6 9 3 9 0 9 4 3 4 4 7 5 3 1 6 1 8
5 0 6 5 6 0 0 1 2 7 6 8 3 6 7 6 6 8 8 2 0 8 1 5 6 8 0 0 1 6 7 8 2 2 4 5 8 3 2 6
8 0 8 8 0 6 3 1 7 1 4 2 8 7 7 6 6 8 3 5 6 0 5 1 5 7 0 2 9 6 5 0 0 2 6 4 5 5 8 7
8 6 4 2 0 4 0 8 5 3 5 3 7 9 8 8 9 4 5 4 6 8 1 3 0 9 1 2 5 3 8 8 1 0 4 7 4 3 1 9
6 0 0 9 7 8 6 4 3 6 0 1 8 6 9 4 7 7 5 8 8 9 5 3 5 9 9 4 0 0 4 8 2 6 8 3 0 6 0 6
5 2 5 8 7 7 1 9 6 5 8 5 4 5 3 4 6 8 3 4 0 0 9 9 1 9 9 7 2 9 7 6 9 4 8 1 5 9 4 1
8 9 1 5 5 9 0 5 5 3 9 0 6 8 9 4 8 6 3 7 0 7 9 5 5 4 7 0 6 2 7 1 1 8 2 6 4 4 9 3
01 Alaska Airlines 11 DuPont 21 Lucent
02 Alcoa 12 Exxon Mobil 22 Mattel
03 Ashland 13 General Dynamics 23 Mead
04 Bank of America 14 General Electric 24 Microsoft
05 BellSouth 15 General Mills 25 Occidental Petroleum
06 Chevron 16 Halliburton 26 JCPenney
07 Citigroup 17 IBM 27 Procter & Gamble
08 Clorox 18 Kellog 28 Ryder
09 Delta Air Lines 19 KMart 29 Sears
10 Disney 20 Lowe’s 30 Time Warner
• The following companies constitute the final sample:

Alaska airlines, Clorox, General mills, Halliburton, kellog
and Procter and Gamble.
2.Stratified Random Sampling…
• Population is divided into non-overlapping
subpopulations called strata.
• A random sample is selected from each

stratum.
• Commonly employed when the population

is heterogeneous or dissimilar, yet similar
homogenous groups can be isolated.
• Example:
Sratum Income categories
1 Under 1000
2 1000-under 5000
3 5000 and above
• If we would like to know whether people in different income categories
differ in their opinions about the proposed tax increase.
• A stratified random sample can be obtained by selecting a simple random

sample of people from each of the 3 income categories.
• Within each Income subgroup (stratum), homogeneity or alikeness is

present; between each pair of subgroups a difference, or heterogeneity, is
present.
•
Stratified by Income
Under 1000
(homogeneous within)
(alike) Heterogeneous
(different)
1000-under 5000 between
(alike) Heterogeneous
(different)
5000 and above between
(alike)
• Stratification is often done by using demographic
variables, such as gender, socioeconomic class,
geographic region, religion,…
• Stratified Sampling reduces the possibility that, by

chance, the sample does not represent the population.
• However, stratified random sampling is generally more

costly than simple random sampling because each unit
of the population must be assigned to a stratum before
the random selection process begins.
• Stratified random sampling can be either
proportionate or disproportionate.
• Proportionate stratified random sampling:
• The percentage of the sample taken from each

stratum is proportionate to the percentage that
each stratum is within the whole population.
• Example:
• Suppose students are being surveyed in FEPS
and the sample is being stratified by gender. If
FEPS population is 70% females and if a sample
of 100 students is taken, the sample would
require inclusion of 70 females and 30 males to
achieve proportionate stratification.
• Any other number of females and males would

be disproportionate stratification.
3.Systematic Random Sampling.
• With systematic sampling, every kth item is selected
to produce a sample of size n from a population of
size N.
• The first sample element is selected randomly from

the first k population elements. Thereafter, sample
elements are selected at a constant interval, k
(sampling cycle), from the ordered sequence frame.
• Convenient and relatively easy to administer.
• The source of population elements should be

random.
3.Systematic Random Sampling…
N
k = ,
n
where:
n = sample size
N = population size
k = size of selection interval
3.Systematic Random Sampling…
• Example:
• Students in a certain class are serialized 1 to 20 (N =
20).
• A sample of 5 (n = 5) students is needed.
• k = 20/5 = 4.
• First sample element is randomly selected from the first

4 students. Assume the 2nd student order was selected
as the first sample element.
• Subsequent sample elements: 6,10,14,18.

Sampling and Non-Sampling
Errors…
• Sampling error
• Nonsampling errors
Sampling Error…
• Sampling error :
• When random sampling techniques are used to select elements

for the sample, sampling error occurs by chance.
• Many times the statistic computed on the sample is not an

accurate estimate of the population parameter because the
sample was not representative of the population.
• Increasing the sample size will reduce this type of error.
• Sampling error will disappear if the whole population is inspected

(census).
Sampling Error…
• Example:
• If we want to determine the mean wage for all workers in the

Egyptian factories.
• The value of the sample mean will deviate from the population
mean because the value of the sample mean depends on
which items were selected for the sample.
• The difference between the true value of the population mean

and its estimate, the sample mean is the sampling error (the
difference between the value of the population parameter and
the value of the corresponding sample statistic).
Nonsampling Error…
• All errors other than sampling errors are nonsampling
errors.
• Increasing the sample size will not reduce this type of

error. Even the census can contain non-sampling errors.
• E.g.: missing data, recording errors, errors resulting

from defective questionnaires, improper definition of the
frame (frame does not fit the population). Response
errors (which occur when people do not know, will not
say, or overstate).
Exercises
• Specify the error type resulting from each of the
following cases:
• An interviewer recording the age of a 24 years old person

as 42.
• Providing an incorrect answer by a respondent due to

misinterpreting an interview question.
Exercises
• Name the sampling plan in each of the following cases.
a. There are four employees' classifications: faculty members,

assistants, administrators, and maintenance workers.
Randomly choose 50 individual from each
category……………………………
b. Each employee has an ID number. Randomly select 200

numbers…………………………………….
c. A list of all names of employees, ordered alphabetically, is

found and every 40th name is chosen to be in the
sample……………………………………….
Exercises
• A city’s telephone book lists 100,000
people. If the telephone book is the frame
for a study, how large would the sample
size be if systematic sampling were done
on every 200th person?

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Sampling and Sampling

• In an experimental study, a variable of interest is first

• Surveys may be administered in a variety of ways, e.g.

• A frame A list, map, directory, or other source used to

• Statistical inference permits us to draw conclusions

• Now, the question is: how a sample will be taken from

• Random Sampling Techniques.

• Sampling and Non-Sampling Errors.

• Random sampling implies that chance enters into the

Nonrandom sampling (Non-probability sampling ):

• Members of nonrandom samples are not selected by

• A non-random sample of homes for door-to-door

• In contrast, a random sample would require the

• Nonrandom sampling methods are not appropriate

• The statistical methods presented and discussed in

• We will focus our attention on three basic

1. Simple Random Sampling.

2. Stratified Random Sampling.

3. Systematic Random Sampling.

• Use a random number table or a random number generator

• Easier to perform for small populations and difficult to apply

• E. g : Assign a number to each student in the class, write

01 Alaska Airlines 11 DuPont 21 Lucent

• The following companies constitute the final sample:

• A random sample is selected from each

• Commonly employed when the population

• A stratified random sample can be obtained by selecting a simple random

• Within each Income subgroup (stratum), homogeneity or alikeness is

• Stratified Sampling reduces the possibility that, by

• However, stratified random sampling is generally more

• Proportionate stratified random sampling:

• The percentage of the sample taken from each

• Any other number of females and males would

• The first sample element is selected randomly from

• Convenient and relatively easy to administer.

• The source of population elements should be

• A sample of 5 (n = 5) students is needed.

• First sample element is randomly selected from the first

• Subsequent sample elements: 6,10,14,18.

• When random sampling techniques are used to select elements

• Many times the statistic computed on the sample is not an

• Increasing the sample size will reduce this type of error.

• Sampling error will disappear if the whole population is inspected

• If we want to determine the mean wage for all workers in the

• The difference between the true value of the population mean

• Increasing the sample size will not reduce this type of

• E.g.: missing data, recording errors, errors resulting

• An interviewer recording the age of a 24 years old person

• Providing an incorrect answer by a respondent due to

a. There are four employees' classifications: faculty members,

b. Each employee has an ID number. Randomly select 200

c. A list of all names of employees, ordered alphabetically, is

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