Solution to Exercise 3.77.

  Let X be the number in the sample of n = 1000 who carry the gene for colon cancer. Assuming the sample is a random sample without replacement (which is usually the understanding in these problems), then X has in fact a hypergeometric distribution. However, we aren't given the population size, so we couldn't compute with the hypergeometric distribution. Furthermore, it is reasonable to assume the population size is much larger than the sample size n = 1000, so the binomial approximation to the hypergeometric applies, and we can take $X \sim Bin(1000,.005)$ (note .005 = 1/200). Now, we use the Poisson approximation to the Binomial as n is large and p is small, so we can take $X \sim Poisson(\lambda = 1000*.005 = 5)$.

(a)

$$P[ 5 \le X \le 8 ] \; = \; P[ X \le 8 ] - P[ X \le 4 ] \; = \; .932 - .440 \; = \; .492 .$$

The numbers come from the table on p. 721.

(b)

$$P[ X \ge 8 ] \; = \; 1 - P[ X \le 7 ] \; = \; 1 - .867 \; = \; .133 .$$