Question

A recent survey shows that 60% of the factory workers willing to work overtime without extra pay as long as they are guaranteed job security and bonus at the end of the year. Fifteen factory workers are randomly selected, find (Use binomial formula):

(a) The mean and standard deviation of the number of workers who are willing to work overtime without extra pay.

(b) The probability that at least 13 workers are willing to work overtime without extra pay.

(c) The probability that 10 workers are unwilling to work overtime without extra pay.

(d) Explain why a binomial distribution is suitable for computing probabilities of the number of workers who are willing to work overtime without pay.

Answer 1

Solution:
p(Success) = 0.6
P(Failure) = 0.4
No. of sample = 15
Solution(a)
mean = n*p = 15*0.6 = 9
Standard deviation = sqrt(npq) = sqrt(15*0.6*0.4) = sqrt(3.6) = 1.897
Solution(b)
P(X>=13) = P(X=13) +P(X=14) + P(X=15) = 15C13*(0.6)^13*(0.4)^2 + 15C14*(0.6)^14*(0.4)^1 + 15C13*(0.6)^13*(0.4)^0 = 0.0219+0.0047+0.00047 = 0.0271

Solution(c)
P(X=10 who are unwilling to work) = 15C10*(0.4)^10 *(0.6)^5 = 0.0245
Solution(d)

Because every worker is independent to each other and there is two type of responses like yes or no for this. so binomial distribution is suitable for computing probabilities of the number of workers who are willing to work overtime without pay.