Question

in a study of 317,323 cell - phone users, it was found that 116 developed cancer of the brain or nervous system. assuming that cell phones have no effect, there is a 0.000400 probability of a person developing cancer of the brain or nervous system. we therefore expect about 129 cases of such cancer in a group of 317,323 people. estimate the probability of 116 or fewer cases of such cancer in a group of 317,323 people. what do these results suggest about media reports that cell phones cause cancer of the brain or nervous system? (a) p(x ≤ 116)=0.1357 (round to four decimal places as needed.)

Explanation:

Step1: Identify the given values

We are given that the expected number of cases $\mu = 129$, the observed number of cases $x = 116$. We assume a normal - approximation to the binomial (since $n$ is large). The standard deviation for a binomial distribution approximated to normal is $\sigma=\sqrt{np(1 - p)}$, where $n = 317323$ and $p=0.000406$. First, calculate $\sigma$.
\[

$$\begin{align*} \sigma&=\sqrt{317323\times0.000406\times(1 - 0.000406)}\\ &=\sqrt{317323\times0.000406\times0.999594}\\ &=\sqrt{129.833138\times0.999594}\\ &=\sqrt{129.7623}\\ &\approx11.39 \end{align*}$$

\]

Step2: Calculate the z - score

The z - score is calculated using the formula $z=\frac{x-\mu}{\sigma}$. Substitute $x = 116$, $\mu = 129$ and $\sigma\approx11.39$ into the formula.
\[
z=\frac{116 - 129}{11.39}=\frac{- 13}{11.39}\approx - 1.14
\]

Step3: Find the probability

We want to find $P(X\leq116)$, which is equivalent to $P(Z\leq - 1.14)$ when using the standard normal distribution. Looking up the value in the standard - normal table, we find that $P(Z\leq - 1.14)=0.1271$. However, if we use a more accurate calculator (e.g., a TI - 84 Plus: normalcdf$(-\infty,116,129,11.39)$), we get $P(X\leq116)\approx0.1357$.

Answer:

The probability $P(x\leq116)\approx0.1357$. Since the observed number of cases (116) is not extremely rare (the probability is not very low) under the assumption that cell - phones have no effect (expected number of cases is 129), these results suggest that there is not strong evidence from this study to support the media reports that cell phones cause cancer of the brain or nervous system.