Question

A total of 500 voters are randomly selected in a certain precinct and asked whether they plan to vote for the Democratic incumbent or the Republican challenger. Of the 500 surveyed, 350 said they would vote for the Democratic incumbent. Using the 0.99 level of confidence, what are the confidence limits for the proportion that plan to vote for the Democratic incumbent

Answer 1

Answer:

The limits are 0.6472 and 0.7528

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 $1 - \frac{\alpha}{2}$.

Of the 500 surveyed, 350 said they would vote for the Democratic incumbent.

This means that $n = 500, \pi = \frac{350}{500} = 0.7$

99% confidence level

So $\alpha = 0.01$, z is the value of Z that has a pvalue of $1 - \frac{0.01}{2} = 0.995$, so $Z = 2.575$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.7 - 2.575\sqrt{\frac{0.7*0.3}{500}} = 0.6472$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.7 + 2.575\sqrt{\frac{0.7*0.3}{500}} = 0.7528$

The limits are 0.6472 and 0.7528