Chapter 6: Q73E (page 364)

Bacteria in bottled water. Is the bottled water you drinksafe? The Natural Resources Defense Council warns thatthe bottled water you are drinking may contain morebacteria and other potentially carcinogenic chemicals thanallowed by state and federal regulations. Of the more than1,000 bottles studied, nearly one-third exceeded governmentlevels (www.nrdc.org). Suppose that the NaturalResources Defense Council wants an updated estimate ofthe population proportion of bottled water that violates atleast one government standard. Determine the sample size(number of bottles) needed to estimate this proportion towithin 0.01 with 99% confidence.

Short Answer

Expert verified

The sample size needed to estimate this proportion to within 0.01 with 99% confidence is 14746

Step by step solution

Given information

The Natural Resources Defense Council warns that the bottled water you are drinking may contain more bacteria and other potentially carcinogenic chemicals than allowed by state and federal regulations. Of the more than 1,000 bottles studied, nearly one-third exceeded government levels

Finding the sample size

Here the sample proportion is 1/3.

Therefore,

$\begin{matrix} \hat{p} = \frac{1}{3} \\ = 0.333333 \end{matrix}$

$\begin{matrix} \hat{q} = 1 - \hat{p} \\ = 1 - 0.333333 \\ = 0.666667 \end{matrix}$

Here the standard error is 0.01

The critical value for a 99% confidence interval is $z_{α / 2} = z_{0.01 / 2} = z_{0.005} = 2.576$

$\begin{matrix} S E = z_{α / 2} \sqrt{\frac{\hat{p} \hat{q}}{n}} \\ n = {z^{2}}_{α / 2} \frac{\hat{p} \hat{q}}{S E^{2}} \\ n = {2.576}^{2} \times \frac{0.333333 \times 0.666667}{{0.01}^{2}} \\ n = 14746.17 \\ n \approx 14746 \end{matrix}$