Question

At 95% confidence, how large a sample should be taken to obtain a margin of error of 0.05 for the estimation of a population proportion

Answer 1

Answer:

A sample of 385 is needed.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

The margin of error is:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}}$

95% confidence level

So $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

How large a sample:

We need a sample of n.

n is found when M = 0.05.

We dont know the true proportion, so we work with the worst case scenario, which is $\pi = 0.5$

$M = z\sqrt{\frac{\pi(1-\pi)}{n}}$

$0.05 = 1.96\sqrt{\frac{0.5*0.5}{n}}$

$0.05\sqrt{n} = 1.96*0.5$

$\sqrt{n} = \frac{1.96*0.5}{0.05}$

$(\sqrt{n})^{2} = (\frac{1.96*0.5}{0.05})^{2}$

$n = 384.16$

Rounding up

A sample of 385 is needed.