# In a simple random sample of 400 voters, 220 said that they were planning to vote for the...

## Question:

In a simple random sample of {eq}400 {/eq} voters, {eq}220 {/eq} said that they were planning to vote for the incumbent mayor in the next election. Construct a {eq}99 \% {/eq} confidence interval for the proportion of voters who plan to vote for the incumbent mayor in the next election.

## Confidence Interval:

Confidence interval gives range of uncertainty true population proportion is most likely to contained below or above the best point estimate (P-hat). The range is determined by margin of error.

Given that;

{eq}n=400\\x=220 {/eq}

Use equation below to construct 99% confidence interval:

{eq}\displaystyle \left(\hat P\pm z_{\frac{\alpha}{2}}\times \sqrt{\frac{p(1-p)}{n}}\right) {/eq}

Find critical value z that correspond to 99% level of confidence:

{eq}\frac{\alpha}{2}=\frac{1-0.99}{2}=0.005\\Z_{0.005}=\pm 2.58 {/eq}

Calculate the p-hat:

{eq}\displaystyle \hat p=\frac{220}{400}=0.55 {/eq}

Plug in values into the formula and calculate upper and lower bounds of 99% confidence interval:

{eq}\displaystyle \left(0.55\pm 2.58\times \sqrt{\frac{0.55(1-0.55)}{400}}\right)\\(0.55\pm 0.06)\\(0.49, 0.61) {/eq} 