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. For the company's sample, the probability that the proportion of people who are in the labor force is greater than 0.65 is ________.
A) 0.1179
B) 0.3000
C) 0.3836
D) 0.6179

Question

Quizplus · Accepted Answer

We can use the normal approximation to the binomial distribution to estimate the probability. 
The sample size n=120 is large enough, so we can assume that the sampling distribution of the sample proportion is approximately normal. 
The mean of the sampling distribution is equal to the population proportion, which is 0.637. 
The standard deviation of the sampling distribution can be estimated as: 
sqrt[(0.637)(1-0.637)/120] = 0.049 
To find the probability that the proportion of people in the labor force is greater than 0.65, we need to standardize this value:
Z = (0.65 - 0.637) / 0.049 = 0.265 
Using a standard normal table or a calculator, we find that the probability of Z being greater than 0.265 is 0.3836. Therefore, the probability that the proportion of people who are in the labor force is greater than 0.65 is 0.3836 or approximately 0.38.

The Labor Force Participation Rate Is the Number of People