Question

According to an article the bottled water you are drinking may contain more bacteria and other potentially carcinogenic chemicals than are allowed by state and federal regulations. Of the more than 1500 bottles​ studied, nearly​ one-third exceeded government levels. Suppose that a department wants an updated estimate of the population proportion of bottled water that violates at least one government standard. Determine the sample size​ (number of​ bottles) needed to estimate this proportion to within plus or minus0.02 with​ 99% confidence.

Answer 1

Answer:  3687

Step-by-step explanation:

Given : Prior proportion of bottles that exceeds government levels = $\dfrac{1}{3}$

Margin of error : $E=0.02$

Significance level : $\alpha: 1-0.99=0.01$

Critical value : $z_{\alpha/2}=2.576$

The formula to find the sample size is given by :-

$n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2$

i.e. $n=\dfrac{1}{3}(1-\dfrac{1}{3})(\dfrac{2.576}{0.02})^2$

$3686.54222222\approx3687$

Hence, the minimum required sample size =3687