Transcript Sample size determinations

Sample size
determinations
(Session 11)
SADC Course in Statistics
Learning Objectives
By the end of this session, you will be able to
• explain how sample size calculations are
done when using simple random sampling
when the main objective is one of
estimating a mean or a proportion
• derive the required sample size from first
principles for a simple scenario
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Traditional view
• As indicated in Session 03, statistical theory
closely follows the objective of needing to
produce estimates of population
characteristics
• Sample size formulae generally relate to this
objective
• Note however, that such formulae relate to
very simple scenarios…
• Here we discuss formulae applicable when the
objective is one of estimating a mean or a
proportion – see Session 12 for more general
issues…
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An example
We use an example to illustrate…
• Consider Example of survey described in
Practical 6, question 2.
• Aim was to determine the mean area of land
per farm, of subsistence farmers, using a
simple random sample of farms
• Suppose for a future survey of a similar
nature, it is required to estimate the mean
area of a farm to within 20 acres of the true
value with more than 95% confidence
• How do we determine the sample size?
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Sample size for estimating a mean
Require sample size n so that
Prob{|X - x| < 20}  0.95
But we know
2
S
x ~ N(X, (1  f)
)
n
Hence we need n so that
 |X - x|

20
Prob 

  0.95
S (1 - f)/n 
 S (1 - f)/n
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Sample size for estimating a mean
Above expression implies we need to have
20
z 
S (1 - f)/n
i.e.
i.e.
1
1  20 



n
N  S(z  ) 
 20 
N-n


nN
 S(1.96) 
2
2
Using N=379, and the previous estimate of
s2=6671, get n54.8. Thus use n=55.
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Formula for sample size: with fpc
The general formula for sample size is:
n 
N
 d 
1 + N

 zS 
2
Incorporates
the finite
population
correction (fpc).
Here d=minimum difference required from
true mean, z is the value from z tables to get
(1- )100% confidence, and S is the
population variance using a suitable estimate.
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Formula for sample size: ignoring fpc
If the finite population correction can be
ignored, this formula simplifies to:
 S2 z 2 
n  

2
 d 
Note that the main difficulty in using this
expression is that it requires an estimate of
the population variance.
See separate handout for some hints on how
this may be overcome.
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Sample size for a proportion
The sample size formula when estimating a
population proportion is:
n =
n0
1
1 + (n0  1)
N
where
n0 

n0
.
n
1 + 0N
P(1  P) P(1-P)
 2 2
V
d /z 
Here P is the population proportion which is
unknown. So require some approximate
knowledge of P, or use P(1-P)=¼ which is its
maximum value.
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References
Lemeshow, S., Hosmer, D.W., Klar, J. and Lwanga,
S.K. (1990) Adequacy of Sample Size in Health
Studies. W.H.O./Wiley, 0 471 92517 9.
SSC (2001) Case Studies of Good Statistical
Practice. Case Study 5 & 8. The University of
Reading Statistical Services Centre Guideline Series
for DFID, available at
http://www.rdg.ac.uk/ssc/workareas/development/c
ase_studies.html
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Practical work follows…
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11