In the following problem check that it is appropriate to use

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. It is estimated that 3.1% of the general population will live past their 90th birthday. In a graduating class of 749 high school seniors, find the following probabilities. (Round your answers to four decimal places.) (a) 15 or more will live beyond their 90th birthday (b) 30 or more will live beyond their 90th birthday (c) between 25 and 35 will live beyond their 90th birthday (d) more than 40 will live beyond their 90th birthday

Solution

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is estimated that 3.5% of the general population will live past their 90th birthday. In a graduating class of 741 high school seniors, find the following probabilities. (Round your answers to four decimal places.)

N = 741, p = .035.

If n*p*(1-p) > or equal to 10, the binomial random variable is approximately normal.

For this problem, 741*.035*(1-.035) = 25.03, which is greater than 10. So it is appropriate to use the normal approximation to the binomial.

The mean will be n*p = 741*.035 = 25.935

The standard deviation will be square root of n*p*(1-p) = square root of 741*.035*(1-.035) = 5.002727

(a) 15 or more will live beyond their 90th birthday

Using the correction to continuity, we want to find the probability that x is greater than 14.5.

Then convert 14.5 to a z score:

Z = (x