The labor force participation rate is the number of people in the labor force divided by the number of people in the country who are of working age and not institutionalized.The BLS reported in February 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com).A marketing company asks 120 working-age people if they either have a job or are looking for a job,or,in other words,whether they are in the labor force.What is the probability that between 60% and 62.5% of those surveyed are members of the labor force?
A) 0.0243
B) 0.1931
C) 0.2005
D) 0.3936

Question

Quizplus · Accepted Answer

We can model the situation using a binomial distribution, where the probability of success (a person being in the labor force) is p = 0.637 (the reported labor force participation rate).

Let X be the number of people out of the 120 surveyed who are in the labor force. Then, X ~ Binomial(120, 0.637).

We want to find the probability that the proportion of people in the labor force (X/120) falls between 0.6 and 0.625:

P(0.6 ≤ X/120 ≤ 0.625)

= P(X ≥ 72 and X ≤ 75)

Using normal approximation to the binomial, we get:

μ = np = 120(0.637) = 76.44 
σ = sqrt(np(1-p)) = sqrt(120(0.637)(1-0.637)) = 3.75

P(72 ≤ X ≤ 75) ≈ P(Z ≤ (75.5 - 76.44)/3.75) - P(Z ≤ (71.5 - 76.44)/3.75)

≈ P(Z ≤ -1.28) - P(Z ≤ -3.31)

≈ 0.1003 - 0.0005

≈ 0.0998

Therefore, the probability that between 60% and 62.5% of those surveyed are members of the labor force is approximately 0.0998, which is closest to option B (0.1931).

The Labor Force Participation Rate Is the Number of People