AP Statistics X-Bar Question

Question

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 3.6% of the general population will live past their 90th birthday. In a graduating class of 745 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

Answer 1

np(1-p) = .036*745*(1-.036) = 25.9 > 10. Therefore, problem approximately follows a normal distribution.

a) mean = .036*745 = 26.82, standard deviation = sqrt(pn(1-p)) = sqrt(.036*745*(1-.036)) = 5.08. Use TI 84 cumulative distribution function with mean 26.28, sd 5.08, upper bound 10⁹⁹, lower bound 15) = .938

b) normalcdf(mean 26.82, sd 5.08, ub 10⁹⁹, lb 30) = .266

c) normalcdf(mean 26.82, sd 5.08, ub 35, lb 25) = .586

d) normalcdf(mean 26.82, sd 5.08, ub 10⁹⁹, lb 40) = .005