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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.

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JeanaShupp

Answer:  3687

Step-by-step explanation:

Given : Prior proportion of bottles that exceeds government levels = [tex]\dfrac{1}{3}[/tex]

Margin of error : [tex]E=0.02[/tex]

Significance level : [tex]\alpha: 1-0.99=0.01[/tex]

Critical value : [tex]z_{\alpha/2}=2.576[/tex]

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

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

i.e. [tex]n=\dfrac{1}{3}(1-\dfrac{1}{3})(\dfrac{2.576}{0.02})^2[/tex]

[tex]3686.54222222\approx3687[/tex]

Hence, the minimum required sample size =3687

The sample size​ (number of​ bottles) needed to estimate the given proportion is; 3687

What is the required sample size?

We are given;

Sample size; n > 1500

Margin of error; E = 0.02

Confidence Level = 99%

Sample proportion; p = 1/3

Formula for margin of error here is;

E = z√(p(1 - p)/n)

Making sample size n the subject gives;

n = (p(1 - p)) * (z²/E²)  

z at 99% Confidence level = 2.576. Thus;

n = (¹/₃(1 - ¹/₃)) * (2.576/0.02)²

n ≈ 3687

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