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# Sample size and finite population correction factor

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The house of representatives has a very controversial bill that will be going to vote. To help formulate his strategy on how to get the bill passed, the chairman of the committee that has been studying the bill wishes to know if there is a division between Democrats and Republicans on the issue or if there is some other basis for the split. To estimate the proportion of Republicans in favor of the bill, how large must a sample of representatives be to estimate the percentage within 3 percentage points with 90% certainty. Assume there are 538 representatives. Use the finite population correction factor and develop your own formula for computing the answer.

https://brainmass.com/statistics/sample-size-determination/sample-size-finite-population-correction-factor-29101

## SOLUTION This solution is FREE courtesy of BrainMass!

See the attached file.
sp=standard error of proportion=square root of (pq/n) square root of ((N-n)/(N-1))
where square root of ((N-n)/(N-1)) is the finite population multiplier
where N is the population size and n is the sample size

confidence interval= 90%
We have to find z corresponding to 5% in each of the two tails. (100%-2x 5%=90%)
Z corresponding to 90% and two tailed test is 1.6449

We have to see that Z* sp < 3%
or sp < 3%/Z
or sp < 0.018238 =3%/1.6449
But
sp=standard error of proportion=square root of (pq/n) square root of ((N-n)/(N-1))

The maximum value of standard error of proportion for any given value of n is when p=q=0.5
pq= 0.25 =0.5*0.5
square root of ((N-n)/(N-1))= square root of ((538-n)/(538-1))=square root of (538-n)/537)
as N= 538

Thus
square root of (pq/n) square root of ((N-n)/(N-1))< 0.018238
Or square root of (0.25/n) square root of ((538-n)/(537))< 0.018238

Or squaring both sides

(0.25/n) (538-n)/(537)< 0.00033262

Or (538-n)/n= 0.714468

Or 538/n -1 = 0.714468

or 538/n= 1.714468
or n= 313.8 which when rounded off into a whole number= 314