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You are here: Experts > Science > Mathematics > Probability & Statistics > Computing Margin of Error
Expert: ronny fisher - 11/2/2009
Question
Hello Ronny,
This is a follow-up to a previous question for which you helped me, thanks again for that.
The formula for MoE you gave was:
ME = 1.96 * sqrt{p x qb / n}
for a proportion statistic of sample size n.
This formula does not seem to depend upon the overall population size from which the sample, of size "n", was drawn.
For proportion statistics, given a confidence level, say 95%, is there a method to determine a MoE that is a function of the population size and the sample size?
From the overall population of 10,000, I have several disjoint subsets that responded to a survey. They are sub-populations and samples size of ...
30 out of 330
10 out of 50
100 out of 800
250 out of 4900
... and similar ratios.
It seems like there'd be various errors based on the sample sizes as a fraction of the population.
I stumbled onto this site:
http://www.raosoft.com/samplesize.html
but I don't see how they are computing the MoEs.
Thanks for your help Ronny.
best,
--dave
Answer
dave -
you are quite correct that the ME should also take into account the
population size. the actual formula for the ME is
ME = 1.96sqrt[(pq/n)(N-n)/N ].
here N is the population size and n is the sample size.
for large N [your N=10,000], the term involving N will usually be very close
to 1 and can be ignored for practical purposes. even if N is not so large,
that term [called the finite population correction - or FPC] can be omitted.
doing so only makes the ME a bit larger - which is being conservative.
it is pretty standard practice in statewide or even municipal surveys, where
N will be 100.000 or more [or even in the millions] to ignore the FPC. but
technically it should be included.
for N=330 and n=30, the FPC= (330-30)/330 = .909 and sqrt{FPC} = .953.
so using the FPC allows you to reduce the ME by about 5%. not a huge
reduction - but better than nothing, i suppose.
you can get this information and much more from the undergrad sampling text:
elementary survey sampling, 6th ed., by r. scheaffer, w. mendenhall and l. ott,
published by duxbury. [an older edition will also have this sort of basic
information.]
ronny
ps: the raosoft website you cite does have the FPC. look at the expression
for what is called 'E' there. [i don't find the notation used there especially
helpful. but at least it's free.]
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