Question

Exposure to a chemical in Teflon-coated cookware. Perfluorooctanoic acid (PFOA) is a chemical used in Teflon-coated cookware to prevent food from sticking. The EPA is investigating the potential risk of PFOA as a cancer-causing agent (Science News Online, August 27, 2005). It is known that the blood concentration of PFOA in people in the general population has a mean of parts per billion (ppb) and a standard deviation of ppb. Science News Online reported on tests for PFOA exposure conducted on a sample of 326 people who live near DuPont’s Teflon-making Washington (West Virginia) Works facility.

a. What is the probability that the average blood concentration of PFOA in the sample is greater than 7.5 ppb?

b. The actual study resulted in $\overline{\mathbf{x}}\mathbf{=}\mathbf{300}$ppb. Use this information to make an inference about the true mean$\mathbf{\mu }$PFOA concentration for the population of people who live near DuPont’s Teflon facility.

Short answer

a. The probability that the average blood concentration of PFOA in the sample is greater than 7.5 ppb is 0.0034.

b. It can be concluded that the true mean $\mu$PFOA concentration for the population of people who live near DuPont’s Teflon facility is not 6 ppb but it is greater than 6ppb because the $P\left(\overline{x}⩾300\right)$is absolutely zero when the population mean is 6 ppb.

Step-by-step solution

Given information 

Science News Online reported on tests for PFOA exposure conducted on a sample of 326 people who live near DuPont’s Teflon-making Washington. So, sample size n=326. Also, it is given that$\mu =6$ and$\sigma =10$

Calculating the probability 

a.

Let X be the blood concentration of PFOA.

From the given problem$\mu =6$ ,$\sigma =10$ and sample size is $n=326$.

According to Central limit theorem, if the sample size is large, then the sampling distribution of the sample mean $\left(\overline{x}\right)$becomes approximately normal.

Let,

$\begin{array}{rcl}P\left(\overline{x}>7.5\right)& =& P\left(\frac{\overline{x}-\mu }{\sigma }>\frac{7.5-6}{10}\right)\\ & =& P\left(Z>\frac{1.5}{0.5538}\right)\\ & =& P\left(Z>2.71\right)\\ & =& 1-P\left(Z⩽2.71\right)\\ & =& 1-\left\{0.5+P\left(0<Z<2.71\right)\right\}\\ & =& 1-0.5-0.4966\\ & =& 0.0034\\ & & \end{array}$

Thus, the probability that the average blood concentration of PFOA in the sample is greater than 7.5 ppb is 0.0034.

Interpretation

b.

Now,

$\begin{array}{rcl}P\left(\overline{x}⩾300\right)& =& P\left(\frac{\overline{x}-\mu }{\sigma }>\frac{300-6}{10}\right)\\ & =& P\left(Z>\frac{294}{0.5538}\right)\\ & =& P\left(Z>530.88\right)\\ & =& 1-P\left(Z<530.88\right)\\ & =& 1-\left\{0.5+P\left(0<Z<530.88\right)\right\}\\ & \cong & 1-0.5-0.5\\ & \cong & 0\\ & & \end{array}$

Thus,$P\left(\overline{x}⩾300\right)\cong 0$

Thus, it can be concluded that the true mean$\mu$ PFOA concentration for the population of people who live near DuPont’s Teflon facility is not 6 ppb but it is greater than 6ppb because the is$P\left(\overline{x}⩾300\right)$ absolutely zero when the population mean is 6 ppb.