# The resuts of a recent statewide test suggested that the proportion is 0.74. Using this estimate, wh

The resuts of a recent statewide test suggested that the proportion is 0.74. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.07?
A sample of 7410 elementary school children is needed to obtain a 99.9% confidence interval with a margin of error of 0.07.

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Shawn Kim
Given: proportion p=0.74
Margin for Error = E = 0.07
Sample size for 9.99% confidence.
$$\displaystyle\alpha={0.001}$$
$$\displaystyle{z}_{{\frac{{x}}{{2}}}}={z}_{{{0.0005}}}={3.29}$$
we know $$\displaystyle{E}={z}_{{\frac{{x}}{{2}}}}\cdot\sqrt{{\frac{{{p}{\left({1}-{p}\right)}}}{{n}}}}$$
So,
$$\displaystyle{n}={p}{\left({1}-{p}\right)}{\left[\frac{{{z}_{{\frac{{x}}{{2}}}}}}{{E}}\right]}^{{2}}$$
$$\displaystyle={0.74}{\left({1}-{0.74}\right)}{\left[\frac{{{3.29}}}{{{0.07}}}\right]}^{{2}}$$
$$\displaystyle={425.01}$$
$$\displaystyle{n}={425}$$
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Ella Williams
A sa,ple of 425 elementary school children is neded to obtain a 9.99% confidence interval with a margin of error of 0.07