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Last modified: January 26, 2018

Sample Size Determination
-------------------------

The error arising due to drawing inferences about the population 
on the basis of sampling is termed as sampling error. The margin 
of error is the maximum error expected in stating, at a level of 
confidence, an interval within which the population parameter is 
supposed to lie. 

Margin of Error, MOE = Zα/2σ/√n
                      _             _
Confidence interval:  x - MOE < μ < x + MOE
     _             _
Take x - MOE = μ ∴ x - μ = MOE

Or, for a proportion:

MOE = Zα/2√(pq/n) 

Rearranging,
          _
n = (Zα/2σ/x - μ)2 

  = (Zα/2σ/MOE)2 

Or, for a proportion:

n = (Zα/2/MOE)2pq  using σ = √(pq/n)
                
In the case where p is unknown, a value of 0.5 is used to provide
a conservative estimate of the sample size.  The approximated MOE 
formula is then:

MOE = ±1.96√(0.5(1 - 0.5)/n) 

    = ±1.96√(0.25/n) 

    = ±1.96*0.5√(1/n) 

    ~ ±1/√n

Ex.

In a random survey of 1,000 Floridians, 43% of the respondents liked 
Ford pickups over Chevy pickups and 52% liked Chevy pickups over 
Ford pickups.  5% had no preference. 

First, set n = 1,000 and p = 0.43. Then

±1.96√((0.43*0.52)/1000) = ±0.0293

This means that if you perform the same survey 100 more times, then 
95% of the time the number of people who liked Ford more than Chevy 
should be between 40.1% and 45.9%.

 

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