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) 0.1926
C) 0.2005
D) 0.3936

Question

Quizplus · Accepted Answer

We can model the situation using a binomial distribution where the probability of success (being a member of the labor force) is p = 0.637 (corresponding to the reported labor force participation rate) and the number of trials (sample size) is n = 120.

We want to find the probability that the proportion of successes (sample proportion of people in the labor force) p̂ falls between 0.6 and 0.625, or equivalently, that 72 ≤ X ≤ 75, where X is the number of "successes" (people in the labor force) in the sample.

Using a continuity correction, we can convert this to a range of values for a standard normal distribution, by subtracting 0.5 from the lower bound and adding 0.5 to the upper bound:

(72 - 0.5 - np) / sqrt(np(1-p)) ≤ Z ≤ (75 + 0.5 - np) / sqrt(np(1-p))

where Z is a standard normal variable, n = 120, and p = 0.637.

Plugging in these values, we get:

(72 - 0.5 - 120*0.637) / sqrt(120*0.637*(1-0.637)) ≤ Z ≤ (75 + 0.5 - 120*0.637) / sqrt(120*0.637*(1-0.637))

-1.89 ≤ Z ≤ -0.96

Using a standard normal table or calculator, we can find the probability that Z falls in this range:

P(-1.89 ≤ Z ≤ -0.96) = 0.1926

Therefore, the probability that between 60% and 62.5% of those surveyed are members of the labor force is 0.1926, which is closest to option B.

The Labor Force Participation Rate Is the Number of People