probability of 76

Question

In a study of

360,141

cell phone​ users, it was found that

76

developed cancer of the brain or nervous system. Assuming that cell phones have no​ effect, there is a

0.000225

probability of a person developing cancer of the brain or nervous system. We therefore expect about

82

cases of such cancer in a group of

360,141

people. Estimate the probability of

76

or fewer cases of such cancer in a group of

360,141

people. What do these results suggest about media reports that cell phones cause cancer of the brain or nervous​ system?

​(a)

​P(x≤76​)=***blank***

​(Round to four decimal places as​ needed.)

Answer 1

There is an exact answer and two approximations. Given that the expected value is "large enough" but not too large, I would expect the approximations to be fairly close.

I use Excel for all calculations; any software should match

Given that the problem mentions the expected number, my guess is that it expected the Poisson approximation to be used.

poisson.dist(76,82,1) = .2757

However, 82 is a bad approximation, as np = 360141*.000225 = 81.03173

Using this value, we get poisson.dist(76,81.03173,1) = .3122

This number becomes relevant as we show the other two results.

Exact uses the binomial distribution.

P(X <= 76) = binom.dist(76,360141,.000225,1) = .3122

This matches the Poisson with the correct expected value.

For the normal approximation, for P(X <= 76, we can use the continuity approximation, P(X <= 76.5)

norm.dist(76.5,360141*.000225,sqrt(360141*.000225*(1-.000225)),1) = .3073. This is close, but not as close as the other two. Note that, without the continuity correction, we would be further off.