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.

Answer and Explanation:

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.


Learn more about this topic:

Loading...
Finding Confidence Intervals for Proportions: Formula & Example

from Statistics 101: Principles of Statistics

Chapter 9 / Lesson 8
8.5K

Related to this Question

Explore our homework questions and answers library