# In a poll, 396 out of 825 New York voters surveyed indicated that they approve of the job...

## Question:

In a poll, {eq}396 {/eq} out of {eq}825 {/eq} New York voters surveyed indicated that they approve of the job DeBlasio has done as mayor. Make a {eq}99\% {/eq} confidence interval for the percentage of all New York voters who would approve of DeBlasio's performance as mayor. Describe the meaning of the interval that you found.

## Confidence Interval for a Proportion:

Confidence interval gives a range of all values likely to be true population proportion at a given level of confidence. The length of the interval is determined by sample size, sample variability and level of confidence.

Given that;

{eq}x=396\\n=825 {/eq}

Use equation below to construct 99% confidence interval:

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

#### Step one:

Calculate the P-hat:

{eq}\displaystyle \hat p=\frac{396}{825}=0.48 {/eq}

#### Step two:

Find critical value z:

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

#### Step three:

Calculate the confidence interval:

{eq}\displaystyle \left(0.48\pm 2.58\times \sqrt{\frac{0.48(1-0.48)}{825}}\right)\\(0.48\pm 0.04)\\(0.44, 0.52) {/eq}

#### Step four:

Make the interpretation:

We are 99% confident that the proportion of all voters who indicated that they approve of the job DeBlasio has done as mayor is contained between 0.44 and 0.52.