Question

In a study of 479,908 cell phone​ users, it was found that 213 developed cancer of the brain or nervous system. Assuming that cell phones have no​ effect, there is a 0.000462 probability of a person developing cancer of the brain or nervous system. We therefore expect about 222 cases of such cancer in a group of 479,908 people. Estimate the probability of 213 or fewer cases of such cancer in a group of 479,908 people. What do these results suggest about media reports that cell phones cause cancer of the brain or nervous​ system?

Answer 1

Answer:

The probability of 213 or fewer cases of such cancer in a group of 479,908 people is 0.2776.

Step-by-step explanation:

We are given that assuming that cell phones have no​ effect, there is a 0.000462 probability of a person developing cancer of the brain or nervous system. We, therefore, expect about 222 cases of such cancer in a group of 479,908 people.

Let $\hat p$ = sample probability of a person developing cancer of the brain or nervous system.

The z-score probability distribution for the sample proportion is given by;

                            Z  =  $\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$  ~ N(0,1)

where, p = population probability of a person developing cancer of the brain or nervous system = 0.000462

            n = sample of cell phone​ users = 479,908

Now, the probability of 213 or fewer cases of such cancer in a group of 479,908 people is given by = P($\hat p$ $\leq$ $\frac{213}{479,908}$ )

     P($\hat p$ $\leq$ 0.000444) = P($\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ $\leq$ $\frac{0.000444-0.000462}{\sqrt{\frac{0.000444(1-0.000444)}{479,908} } }$ ) = P(Z $\leq$ -0.59)

                                 = 1 - P(Z < 0.59) = 1 - 0.7224 = 0.2776

The above probability is calculated by looking at the value of x = 0.59 in the z table which has an area of 0.7224.

These results suggest about media reports that cell phones cause cancer of the brain or nervous​ system that there is around 28% chance that 213 or fewer cases of such cancer in a group of 479,908 people can take place.