Question 1191258
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The margin of error for the confidence interval of a population proportion p is
E = z*sqrt(phat(1-phat)/n)
where z is the critical value, phat is the sample proportion, and n is the sample size.
The term "phat" is often written as "p hat" or "p-hat", but I'm deciding to use a shorter version for the sake of compactness.
phat estimates the population proportion p.


Solving for n gets us
E = z*sqrt(phat(1-phat)/n)
E/z = sqrt(phat(1-phat)/n)
(E/z)^2 = phat(1-phat)/n
n*(E/z)^2 = phat(1-phat)
n = phat(1-phat)*(z/E)^2


Since we estimate p = 0.37, this will be plugged into each phat.
At 99% confidence, the z critical value is roughly z = 2.576 (use a calculator or table to find this value).
The desired error we want is E = 0.04


With all that in mind, we can then compute the minimum sample size n
n = phat(1-phat)*(z/E)^2
n = 0.37(1-0.37)*(2.576/0.04)^2
n = 966.749616
We round up to n = 967
We always round up regardless how close or far the decimal value is to the nearest largest integer.
For example, if we got the result n = 966.00001, we would still round up to n = 967
This rounding up is to ensure we clear the hurdle needed to form the min sample size.



Answer: <font color=red>967</font>

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