Question

Election polling Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. We want to estimate the proportion $p$of all registered voters in the city who plan to vote for Chavez with $95$% confidence and a margin of error no greater than $0.03$. How large a random sample do we need? Show your work.

Short answer

From the given information, the sample size of the registered voter is$1068$

Step-by-step solution

Given Information

It is given in the question, the margin of error$E=0.03$

confidence interval$95\%$

How large a random sample do we need?

Explanation

Confidence interval is $95\%$

Convert $95\%$into decimal.

$\frac{95}{100}=0.95$

For confidence interval $0.95,$use table A.

$z_{\alpha /2}=1.96$

Since the sample proportion $\stackrel{\wedge }{p}$is unknown.

Calculate sample size. Use the formula $n=\frac{{\left[z_{\alpha /2}\right]}^2\times 0.25}{E^2}$

$n=\frac{{\left[z_{\alpha /2}\right]}^2\times 0.25}{E^2}$

$n=\frac{{1.96}^2\times 0.25}{{0.03}^2}$

$=1068$

Hence, the required sample size is$1068$.