Question

Factory Farming Funk. During one year, the U.S. Environmental Protection Agency reported that concentrated animal feeding

operations (CAFOs) dump $2$ trillion pounds of waste into the environment annually, contaminating the ground water in $17$ states and polluting more than $35,000$ miles of our nation's rivers. In a survey of $1000$ registered voters by Snell, Perry and Associates, $80\%$ favored the creation of standards to limit such pollution and, in general, viewed CAFOs unfavorably. Find and interpret a $99\%$ confidence interval for the percentage of all registered voters who favor the creation of standards on CAFO pollution and, in general, view CAFOs.

Short answer

There is a $99\%$chance that between $76.7\%$and $83.3\%$of U.S. people dislike CAFOs and support the creation of CAFO pollution standards.

Step-by-step solution

Given Information

Sample size is $n=1000$.

Sample proportion is $\hat{p}=80\%$

$=0.8$.

Explanation

With a confidence interval,

$1-\alpha$, find $z_{\alpha /2}$using table II.

The confidence level is $99\%$.

i.e. $\alpha =0.01$.

Using table II,

$z_{\alpha /2}=z_{0.01/2}$

$=z_{0.005}$

$=2.58$

The range of confidence for localid="1651462567971" $p$is given as-$\hat{p}-z_{\alpha /2}·\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\text{to}\hat{p}+z_{\alpha /2}·\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$\Rightarrow 0.8-2.58\sqrt{\frac{0.8(1-0.8)}{1000}}\text{to}0.8+2.58\sqrt{\frac{0.8(1-0.08)}{1000}}$$\Rightarrow 0.8-0.033\text{to}0.8-0.033$

$\Rightarrow 0.767\text{to}0.833$

$\Rightarrow 76.7\%\text{to}83.3\%$