Question

The Bureau of Alcohol, Tobacco, and Firearms (BATF) has been concerned about lead levels in California wines. In a previous testing of wine specimens, lead levels ranging from 50 to 700 parts per billion were recorded. How many wine specimens should be tested if the BATF wishes to estimate the true mean lead level for California wines to within 10 parts per billion with 95% confidence?

Answer 1

Answer:  1015

Step-by-step explanation:

Given :In a previous testing of wine specimens, lead levels ranging from 50 to 700 parts per billion were recorded.

Here , Range = 700-50=650    [∵ Range = Max-Min]

According to the thumb rule , the range is approximately 4 times the standard deviation.

Let $\sigma$ be the standard deviation, then

$\text{Range}\approx4\times\sigma\\\\\Rightarrow\ \sigma\approx\dfrac{\text{Range}}{4}=\dfrac{650}{4}=162.5$

Thus, $\sigma\approx162.5$

For confidence interval of 95% , the critical z value = $z_{\alpha/2}=1.96$

Formula to find the sample size : $(\dfrac{z_{\alpha/2}\ \sigma}{E})^2$

Now, for the margin of error of E=$\pm 10$, we have

$(\dfrac{1.96\times162.5}{10})^2=1014.4225\approx1015$

Hence, the required sample size = 1015