Chapter 11: Q.11.53 (page 460)

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

Expert verified

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,

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

The confidence level is $99 %$ .

i.e. $α = 0.01$ .

Using table II,

$z_{α / 2} = z_{0.01 / 2}$

$= z_{0.005}$

$= 2.58$

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

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

$\Rightarrow 0.767 to 0.833$

$\Rightarrow 76.7 % to 83.3 %$

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