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Sample Size Determination
Determining Sample Size for
Probability Samples
• Determining sample size for probability
samples involves financial, statistical, and
managerial considerations.
• The larger the sample, the smaller the
sampling error. In turn, the cost of the
research grows with the size of sample.
Relationship Between Sample Size
and Error
3
Determining Sample Size for
Probability Samples
• There are several methods for determining
sample size.
1. To base the decision on the funds available. In
essence, sample size is determined by the
budget. Although seemingly unscientific, this
approach is often a realistic one in the world of
marketing research.
Determining Sample Size for
Probability Samples
• There are several methods for determining
sample size.
2. The so-called rule of thumb method, which
involves determining the sample size based on a
gut feeling or common practice. The client often
requests samples of 300, 400, or 500 in a
request for proposal (RFP).
Determining Sample Size for
Probability Samples
• There are several methods for determining
sample size.
3. A third technique is based on the number of
subgroups to be analyzed. Generally speaking,
the more subgroups that need to be analyzed,
the larger the required total sample size.
Determining Sample Size for
Probability Samples
• In addition to these methods, there are a number
of statistical formulas for determining sample
size.
• Three pieces of data are required to make
sample size calculation:
– an estimate of the population standard deviation,
– the level of sampling error that researcher/client is
willing to accept,
– and the desired level of confidence that the sample
result will fall within a certain range of the true
population value.
Normal Distribution
• The normal distribution is crucial to statistical
sampling theory. It is the bell-shaped and is
symmetric about its mean; the mean, median,
and mode are equal.
Standard Normal Distribution
• The standard normal distribution has the
features of a normal distribution; however,
the mean of the standard normal distribution
is always equal to zero, and the standard
deviation is always equal to one.
Determining Sample Size
• Problems involving means
For example: the task estimating how many
times the average fast-food restaurant user
visits a fast-food restaurant in an average
month.
Sample Size Formula:
Problems involving means
zs 

n 
E
2
where:
n = sample size
z = confidence interval in standard error units
s = standard error of the mean
E = acceptable magnitude of error
Example #1
Suppose a survey researcher, studying
expenditures on lipstick, wishes to have a 95%
confident level (Z) and a range of error (E) of
less than $2.00. The estimate of the standard
deviation is $29.00.
(z is the z-score, e.g. 1.645 for a 90% confidence interval, 1.96 for
a 95% confidence interval, 2.58 for a 99% confidence interval)
Calculation: Example #1
 zs 
n  
E
2
 1.9629.00 


2.00


2
2
 56.84 
2




28
.
42

 2.00 
 808
Example #2
Suppose, in the same example as the one
before, the range of error (E) is acceptable at
$4.00. By how much is sample size is
reduced?
Calculation: Example #2
 zs 
 1.9629.00
n    

4.00 
E

2
2
2
56.84
2




14
.
21

 4.00 
 202
Calculating Sample Size
99% Confidence
(2.57)( 29)

n 

2


 74.53


 2 
2
 [37.265]
 1389
2
2
(2.57)( 29)

n 

4


2
74.53



4


2
 [18.6325]
 347
2
Determining Sample Size
• Problems involving proportions
For example: the problem of estimating the
proportion/percentage of all adults who have
accessed Twitter in the past 90 days. The goal
is to take a simple random sample from the
population of all adults to estimate this
proportion.
Sample Size Formula:
Problems involving proportions
2
Z pq
n
E
2
Where:
z 2pq
n
2
E
n = number of items in samples
Z2 = square of confidence interval in standard error units
p = estimated proportion of success
q = (1-p) or estimated the proportion of failures
E2 = square of maximum allowance for error between true
proportion and sample proportion, or zsp squared.
Calculating Sample Size
at the 95% Confidence Level
p  .6
q  .4
(1. 96 )2(. 6 )(. 4 )
n
( . 035 )2
( 3. 8416 )(. 24 )
001225
. 922

. 001225
 753
