Stats 227 Problem

Question

19. Government workers: The Bureau of Labor Statistics reported that 16% of U.S. nonfarm workers are government employees. A random sample of 50 workers is drawn.

a. Is it appropriate to use the normal approximation to find the

probability that less than 20% of the individuals in the sample are government employees? If so, find the probability. If not, explain why not.

b. A new sample of 90 workers is chosen. Find the probability that more than 20% of workers in this sample are government employees.

c. Find the probability that the proportion of workers in the sample of 90 who are government employees is between 0.15 and 0.18.

d. Find the probability that less than 25% of workers in the sample of 90 are government employees.

e. Would it be unusual if the proportion of government employees in the sample of 90 was greater than 0.25?

Answer 1

a. can use normal approximation if np and n(1-p) > 10. Here n = 50, p = 0.16. np = 8, so not appropriate to use normal approximation

b. n = 90 np = 14.4 so can use normal approximation.

z = (phat - p)/sqrt(p*(1-p)/n)

z = (0.2 - 0.16)/sqrt(0.16*0.84/90) = 1.04

P(p > 20) = P(z > 1.04) = 1 - P(z < 1.04) = 1 - 0.8508 = 0.1492

c. P( 0.15 < p < 0.18)

convert endpoints to z-score:

z = (0.15-0.16)/sqrt(0.16*0.84/90) = -0.26

z= (0.18-0.16)/sqrt(0.16*0.84/90) = 0.52

P(0.15 < p < 0.18) = P(-0.26 < z < 0.52) = P(z < 0.52) - P(z < -0.26) = 0.6985 - 0.3974 - 0.3011

d. P(p < 0.25)

z= (0.25 - 0.16)/sqrt(0.16*0.84/90) = 2.33

P(p < 0.25) = P(z < 2.33) = 0.9901

e. P(p > 0.25) = 1 - 0.9901 = 0.0099, so that would be very