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Sampling Fundamentals 2
Sampling Process
Identify Target Population
Determine Sampling Frame
Select Sampling Procedure
Determine Sample Size
Determining Sample Size – Ad Hoc Methods
• Rule of thumb
– Each group should have at least 100 respondents
– Each sub-group should have 20 – 50 respondents
• Budget constraints
– The question then is whether the study can be
modified or cancelled
• Comparable studies
– Find similar studies and use their sample sizes as
guides
Factors determining sample size
• Number of groups and sub-groups in the sample
that are to be analyzed
• Value of the study and accuracy required
• Cost of generating the sample
• Variability in the population
Revisit definitions
• Mean: the arithmetic average of scores on a
variable
– Only interval / ratio level data
– Categorical data - Mode
• Variance: the average value of the dispersion
(spread) of squared scores from the mean on a
variable. Based on how a response differs from
the average response
• Standard deviation: Square root of the variance
Basic Statistics
Mean
Variance
Standard Deviation
Sample Size
Population

2

N
1 n
C = S Ci
n i =1
n
1
2
2
=
S
C
C
s
(
)
i
n - 1 i =1
Sample
X
s2
s
n
Sample size determination
• In most MR problems we are interested in
knowing the mean. (e.g. mean attitude scores,
mean sales, etc.).
• We want an good estimate of the population
mean
• Since the population mean is generally unknown,
we must select the sample with care so that the
sample mean will be the closest approximation to
the population mean
Sample size determination
• We want a sample that is
– Selected through random sampling
– Is as large as possible
1. Normal Distribution
The entire area under
the curve adds up to 100%
1. Features of normal distributions
•
•
•
•
•
68% of responses between  + 1 
95% of responses between  + 2 
99.99% of responses between  + 3 
Bell shaped curve
Mean = Median = Mode
2. Sampling distribution of means
• Distribution of mean responses on an item, from every
probability sample taken from the same population, the
sample being taken an infinite number of times
• Smaller sample size = unstable means and greater
variability (higher standard error of the mean) and
greater sampling error
• Larger sample sizes = stable means and lower variability
(lower standard error of the mean) and smaller sampling
error
• Sampling distribution of means with larger sample sizes
give a better approximation to the normal distribution
• E(X bar) = µ
3. Standard Error of the Mean
• Standard deviation of the sampling distribution of
means
 xbar = x / n
• I.e. standard error of the mean will equal the
standard deviation of the population divided by
the square root of the sample size
– I.e. the greater the n, the smaller the standard error
• Therefore random sampling with a larger sample
size gives a more accurate estimate
4. Sampling Error
• OS – TS = SE + NSE
• Assuming full care is taken in the research
process, NSE = 0
• Therefore OS – TS = SE
• i.e. Sampling Error is the difference between
True Score and Observed Score
• To minimize SE – larger sample
• To minimize cost – smaller sample
• To trade off we specify SE as a percentage (e.g.
5%, 3%, 10%, etc.)
5. Interval Estimation of population mean
• OS – TS = SE
• i.e. X
bar
-  = SE
•X
bar
varies from sample to sample
•X
bar
+ SE = interval estimate of 
•X
bar
+ z x / n = interval estimate of 
– n = sample size
– z = z value at chosen confidence level
5. Size of Interval Estimate
• Confidence level (e.g. 90%, 95%, 99%, etc.)
– The number of times the population mean must fall
within the confidence interval after repeated samplings
– Lower confidence levels mean smaller sample sizes
and smaller intervals; Higher confidence levels mean
larger sample sizes and larger intervals
• Population standard deviation
– Generally unknown
– Estimated from a previous study, a pilot, judgment or
a worst case scenario
6. Statistical Sample Size
• Specify
– Size of the sampling error that is desired
– Confidence level
– Population standard deviation
•X
bar
+ SE = interval estimate of 
•X
bar
+ z x / n = interval estimate of 
• SE = z x / n
• Therefore, n = Z2 2 /(SE)2