Question

A political candidate has asked you to conduct a poll to determine what percentage of people support her. If the candidate only wants a 3% margin of error at a 90% confidence level, what size of sample is needed?

Answer 1

Answer:

At sample size of 752 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:

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

90% confidence level

So $\alpha = 0.1$, z is the value of Z that has a pvalue of $1 - \frac{0.1}{2} = 0.95$, so $Z = 1.645$.

What size of sample is needed?

A sample size of n is needed.

n is found when M = 0.03.

We do not have a estimate for the true proportion, so we use $\pi = 0.5$, which is when we are going to need the largest sample size.

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

$0.03 = 1.645\sqrt{\frac{0.5*0.5}{n}}$

$0.03\sqrt{n} = 1.645*0.5$

$\sqrt{n} = \frac{1.645*0.5}{0.03}$

$(\sqrt{n})^{2} = (\frac{1.645*0.5}{0.03})^{2}$

$n = 751.67$

Rounding up

At sample size of 752 is needed.