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Determining the Size of
a Sample
Sample Accuracy
• Sample accuracy: refers to how close
a random sample‟s statistic is to the
true population‟s value it represents
• Important points:
– Sample size is not related to
representativeness
– Sample size is related to accuracy
Ch 13 2
Sample Size and Accuracy
• Intuition: Which is more accurate: a
large probability sample or a small
probability sample?
• The larger a probability sample is, the
more accurate it is (less sample
error).
Ch 13 3
A Picture Says 1,000 Words
±
Sample Size and Accuracy
16%
14% n 550 - 2000 = 1,450
12%
4% - 2% = ±2%
Accuracy
10%
8%
6%
4%
2%
0%
1100
1250
1400
1550
1700
1850
2000
50
200
350
500
650
800
950
Sample Size
Probability sample accuracy (error) can be calculated with a
Ch 13 simple formula, and expressed as a ± % number. 4
How to Interpret Sample
Accuracy
• From a report…
– The sample is accurate ± 7% at
the 95% level of confidence…
• From a news article
– The accuracy of this survey is ±
7%…
Ch 13 5
How to Interpret Sample
Accuracy
• Interpretation
– Finding: 60% are aware of our brand
– So… between 53% (60%-7%) and
67% (60%+7%) of the entire
population is aware of our brand
Ch 13 6
Sample Size Axioms
• To properly understand how to
determine sample size, it helps to
understand the following axioms…
Ch 13 7
Sample Size Axioms
• The only perfectly accurate sample is a
census.
• A probability sample will always have
some inaccuracy (sample error).
• The larger a probability sample is, the
more accurate it is (less sample error).
• Probability sample accuracy (error) can
be calculated with a simple formula,
and expressed as a +- % number.
Ch 13 8
Sample Size Axioms
• You can take any finding in the
survey, replicate the survey with the
same probability sample size, and
you will be “very likely” to find the
same finding within the +- range of
the original finding.
• In almost all cases, the accuracy
(sample error) of a probability sample
is independent of the size of the
Ch 13 population. 9
Sample Size Axioms
• A probability sample can be a very
tiny percentage of the population size
and still be very accurate (have little
sample error).
Ch 13 10
Sample Size and
Population Size
• Where is N (size of the population) in
the sample size determination formula?
Population e=±3% Sample e=±4% Sample
Size Size Size
10,000 ____
1,067 ____
600
100,000 ____
1,067 ____
600
1,000,000 ____
1,067 ____
600
100,000,00 ____
1,067 ____
600
0
In almost all cases, the accuracy (sample error) of a
Ch probability sample is independent of the size of the
13 11
population.
Sample Size Axiom
• The size of the probability sample
depends on the client‟s desired
accuracy (acceptable sample error)
balanced against the cost of data
collection for that sample size.
Ch 13 12
Putting It All Together
• MR – What level accuracy do you want?
• MM – I don’t have a clue. of a probability
The size
• MR – National opinion polls use 3.5%.
sample depends on the client’s
• MM – Sounds good to me.
desired accuracy (acceptable
sample error) balanced against
• MR – Okay, that meanscost ofneed a sample of
the we data collection for
1,200. that sample size.
• MM – Gee Whiz. That small?
• MR – Yup, and at a cost of $20 per completion, it
will be $24,000.
• MM – Holy Cow! That much?
• MR – I could do 500 for $10,000, and that would
be 4.4% accurate, or 300 for $6,000 at 5.7%.
• MM – 500 sounds good to me.
Ch 13 13
• There is only one method of
determining sample size that allows
the researcher to PREDETERMINE
the accuracy of the sample results…
The Confidence
Interval Method of
Determining Sample
Size
Ch 13 14
The Confidence Interval Method
of Determining Sample Size
• This method is based upon the
Confidence Interval and the Central
Limit Theorem…
• Confidence interval: range whose
endpoints define a certain percentage
of the response to a question
Ch 13 15
The Confidence Interval Method
of Determining Sample Size
• Confidence interval approach: applies
the concepts of accuracy, variability,
and confidence interval to create a
“correct” sample size
• Two types of error:
– Nonsampling error: pertains to all
sources of error other than sample
selection method and sample size
– Sampling error: involves sample
selection and sample size
Ch 13 16
The Confidence Interval Method
of Determining Sample Size
• Sample error formula:
Ch 13 17
The Confidence Interval Method
of Determining Sample Size
• The relationship between sample size
and sample error:
Ch 13 18
Computations Help Page
1.96
pq 50 times 50
ez
n
Let’s try 3 n’s
1000
Answers this way…
500
Ch 13 19
100
And the answers are…
1.96
pq 50 times 50
ez
n
Let’s try 3 n’s
1000 ±3.1%
500 ±4.4%
Ch 13 20
100 ±9.8%
Review: What does sample
accuracy mean?
• 95% Accuracy
– Calculate your sample‟s finding, p%
– Calculate your sample‟s accuracy, ±
e%
– You will be 95% confident that the
population percentage (π) lies
between p% ± e%
Ch 13 21
Review: What does sample
accuracy mean?
• Example
– Sample size of 1,000
– Finding: 40% of respondents like
our brand
– Sample accuracy is ± 3% (via our
formula)
– So 37% - 43% like our brand
Ch 13 22
The Confidence Interval Method
of Determining Sample Size
• Variability: refers to how similar or
dissimilar responses are to a given
question
• P: percent
• Q: 100%-P
• Important point: the more variability in
the population being studied, the
higher the sample size needed to
achieve a stated level of accuracy.
Ch 13 23
• With nominal data (i.e. yes, no), we
can conceptualize variability with bar
charts…the highest variability is
50/50
Ch 13 24
Confidence Interval Approach
• The confidence interval approach is
based upon the normal curve
distribution.
• We can use the normal distribution
because of the CENTRAL LIMITS
THEOREM…regardless of the shape
of the population‟s distribution, the
distribution of samples (of n at least
=30) drawn from that population will
form a normal distribution.
Ch 13 25
Central Limits Theorem
• The central limits theorem allows us
to use the logic of the normal curve
distribution.
• Since 95% of samples drawn from a
population will fall + or – 1.96 x
sample error (this logic is based upon
our understanding of the normal
curve) we can make the following
statement…
Ch 13 26
• If we conducted our study over and
over, 1,000 times, we would expect
our result to fall within a known
range. Based upon this, we say that
we are 95% confident that the true
population range value falls within
this range.
Ch 13 27
The Confidence Interval Method
of Determining Sample Size
• 1.96 x s.d. defines the endpoints of
the distribution.
Ch 13 28
• We also know that, given the amount
of variability in the population, the
sample size will affect the size of the
confidence interval.
Ch 13 29
So, what have we learned thus
far?
• There is a relationship between:
– The level of confidence we wish to
have that our results would be repeated
within some known range if we were to
conduct the study again, and…
– Variability in the population and…
– The amount of acceptable sample error
(desired accuracy) we wish to have
and…
Ch 13 – The size of the sample! 30
Sample Size Formula
• Fortunately, statisticians have given us a
formula which is based upon these
relationships.
– The formula requires that we
• Specify the amount of confidence we wish
• Estimate the variance in the population
• Specify the amount of desired accuracy
we want.
– When we specify the above, the formula
tells us what sample we need to use…n
Ch 13 31
Sample Size Formula
• Standard sample size formula for
estimating a percentage:
Ch 13 32
Practical Considerations in
Sample Size Determination
• How to estimate variability (p times q)
in the population
– Expect the worst cast (p=50; q=50)
– Estimate variability: Previous
studies? Conduct a pilot study?
Ch 13 33
Practical Considerations in
Sample Size Determination
• How to determine the amount of
desired sample error
– Researchers should work with
managers to make this decision.
How much error is the manager
willing to tolerate?
– Convention is + or – 5%
– The more important the decision,
the more (smaller number) the
Ch 13 34
sample error.
Practical Considerations in
Sample Size Determination
• How to decide on the level of
confidence desired
– Researchers should work with
managers to make this decision.
The more confidence, the larger
the sample size.
– Convention is 95% (z=1.96)
– The more important the decision,
the more likely the manager will
want more confidence. 99%
Ch 13 35
confidence, z=2.58.
Example
Estimating a Percentage in the Population
• What is the required sample size?
– Five years ago a survey showed that 42%
of consumers were aware of the
company‟s brand (Consumers were either
“aware” or “not aware”)
– After an intense ad campaign,
management wants to conduct another
survey and they want to be 95% confident
that the survey estimate will be within
±5% of the true percentage of “aware”
consumers in the population.
– What is n?
Ch 13 36
Estimating a Percentage:
What is n?
• Z=1.96 (95% confidence)
• p=42
• q=100-p=58
• e=5
• What is n?
Ch 13 37
Estimating a Percentage:
What is n?
n=374
• What does this mean?
– It means that if we use a sample size of
374, after the survey, we can say the
following of the results: (assume results
show that 55% are aware)
– “Our most likely estimate of the
percentage of consumers that are „aware‟
of our brand name is 55%. In addition, we
are 95% confident that the true
percentage of „aware‟ customers in the
Ch 13 population falls between 50% and 60%.” 38
Estimating a Mean
• Estimating a mean requires a
different formula (See MRI 13.2, p.
378)
• Z is determined the same way (1.96 or
2.58)
• E is expressed in terms of the units we are
estimating (i.e., if we are measuring
attitudes on a 1-7 scale, we may want
error to be no more than ± .5 scale units
• S is a little more difficult to estimate…
Ch 13 39
Estimating s
• Since we are estimating a mean, we
can assume that our data are either
interval or ratio. When we have
interval or ratio data, the standard
deviation, s, may be used as a
measure of variance.
Ch 13 40
Estimating s
• How to estimate s?
– Use standard deviation from a
previous study on the target
population.
– Conduct a pilot study of a few
members of the target population and
calculate s.
– Estimate the range the value you are
estimating can take on (minimum and
maximum value) and divide the range
Ch 13 41
by 6.
Estimating s
– Why divide the range by 6?
• The range covers the entire
distribution and ± 3 (or 6) standard
deviations cover 99.9% of the area
under the normal curve. Since we
are estimating one standard
deviation, we divide the range by 6.
Ch 13 42
Example
Estimating the Mean of a Population
• What is the required sample size?
– Management wants to know customers‟
level of satisfaction with their service.
They propose conducting a survey and
asking for satisfaction on a scale from 1
to 10. (since there are 10 possible
answers, the range=10).
– Management wants to be 99% confident
in the results and they do not want the
allowed error to be more than ±.5 scale
points.
Ch 13 – What is n? 43
Estimating a Mean:
What is n?
• S=10/6 or 1.7
• Z=2.58 (99% confidence)
• e=.5 scale points
• What is n?
Ch 13 44
Estimating a Mean:
What is n?
n=77
• What does this mean?
– After the survey, management may make
the following statement: (assume
satisfaction mean is 7.3)
– “Our most likely estimate of the level of
consumer satisfaction is 7.3 on a 10-point
scale. In addition, we are 99% confident
that the true level of satisfaction in our
consumer population falls between 6.8
Ch 13 and 7.8 on a 10-point scale” 45
Other Methods of Sample Size
Determination
• Arbitrary “percentage of thumb”
sample size:
– Arbitrary sample size approaches
rely on erroneous rules of thumb.
– Arbitrary sample sizes are simple
and easy to apply, but they are
neither efficient nor economical.
Ch 13 46
Other Methods of Sample Size
Determination
• Conventional sample size specification:
– Conventional approach follows some
convention: or number believed
somehow to be the right sample size.
– Using conventional sample size can
result in a sample that may be too
large or too small.
– Conventional sample sizes ignore the
special circumstances of the survey
Ch 13 47
at hand.
Other Methods of Sample Size
Determination
• Statistical analysis requirements of
sample size specification:
– Sometimes the researcher‟s desire
to use particular statistical technique
influences sample size
Ch 13 48
Other Methods of Sample Size
Determination
• Cost basis of sample size
specification:
– “All you can afford” method
– Instead of the value of the
information to be gained from the
survey being primary consideration
in the sample size, the sample size
is determined by budget factors that
usually ignore the value of the
Ch 13
survey‟s results to management. 49
Special Sample Size
Determination Situations
• Sampling from small populations:
– Small population: sample exceeds
5% of total population size
– Finite multiplier: adjustment factor for
sample size formula
– Appropriate use of the finite
multiplier formula will reduce a
calculated sample size and save
money when performing research on
small populations.
Ch 13 50
Special Sample Size
Determination Situations
• Sample size using nonprobability
sampling:
– When using nonprobability
sampling, sample size is
unrelated to accuracy, so cost-
benefit considerations must be
used.
Ch 13 51
Practice Examples
• We will do some examples from the
questions and exercises at the end of
the chapter on sample size…question
5 on page 386.
Ch 13 52
Practice Examples
• 5a. Using the formula provided in
your text, determine the approximate
sample sizes for each of the following
cases, all with precision (allowable
error) of ±5%: n= z
2
(pq)
2
e
– Variability of 30%, =
2
1. 96 (30 x 70)
2
5
confidence level 3.84 x 2100
=
of 95% 25
8064
=
25
Ch 13 = 322.6 (323) 53
Practice Examples
• 5b. Using the formula provided in
your text, determine the approximate
sample sizes for each of the following
cases, all with precision (allowable
error) of ±5%: n= z
2
(pq)
2
– Variability of 60%,
e
2
2. 58 (60 x 40)
=
confidence level 5
6.66 x 2400
2
=
of 99% 25
15,984
=
25
Ch 13 = 639.4 (639) 54
Practice Examples
• 5c. Using the formula provided in
your text, determine the approximate
sample sizes for each of the following
cases, all with precision (allowable
error) of ±5%: n= z
(pq)2
2
– Unknown variability, e
2
1.96 (50 x 50)
=
confidence level 5
2
3.84 x 2500
of 95% =
25
9600
=
25
Ch 13 55
= 384
Practice Example
• A client wants to survey out-shopping
intentions (percentage of people
saying “yes” to a question regarding
their intentions to out-shop) among
heads of households in Antigonish.
The client wants a ± 3%, 19 times out
of 20. There are 3,000 households in
the catchment area. What sample
size should be used?
Ch 13 56
Continued
• If you expect an incidence rate of
80% and a refusal rate of 50%, how
many surveys should be sent out?
Ch 13 57
