Margin of Error

May 2010
I've been agonizing over this problem involving a sample mean and margin of error. I feel like the sample size I got as my answer is too small?

Question is:

How large a sample is needed to estimate a population proportion to within a margin of error of 4 % at the 98 % confidence level?

First, I calculated the critical value as 2.33 and then plugged the margin of error, .4, and 2.33 into the formula:

n = (z* divided by E)^2 x p-hat x q-hat or (2.33 divided by .4) ^2 x 1/2 x 1/2

I got the 1/2 since p-hat and q-hat are both unknowns so you have to use the most conservative estimate, or 1/2.

Anyway, my answer is 8.49 or sample size of 9

I feel like this is way too small of a sample size! What'd I do wrong?
Apr 2009
Since we have that \(\displaystyle (1-\alpha)=.98\), \(\displaystyle \alpha\) must equal 0.02 and \(\displaystyle \frac{\alpha}{2}=.01\). So, the z value we are looking for is \(\displaystyle z_{\frac{\alpha}{2}}=z_{.01}=2.33\). We then require that

\(\displaystyle 2.33\sqrt{\frac{pq}{n}}=.4\).

Since the we do not know anything about \(\displaystyle p\) we use \(\displaystyle p=.5\).

I have not computed the value, but if you just solve for n you should find your answer. However, I think your answer is correct.