Question

The technical area of DISTELSA has 100 workers and it has been observed that the time they spend repairing a particular electronic equipment has a normal distribution with an average of 80 minutes and a standard deviation of 8 minutes.
a) Calculate the probability that a randomly chosen worker can repair equipment in less than an hour and 15 minutes.
b) Calculate the probability that a randomly chosen worker could take between 80 minutes and one hour ten minutes
c) What is the probability of exactly 20 of the 100 workers, it takes between 80 minutes and one hour ten minutes?

Answer 1

X: Time to repair equipment

X has normal distribution with mean 80 minutes and standard deviation 8 minutes

a)

probability that a randomly chosen worker can repair equipment in less than an hour and 15 minutes (75 miutes)

= P(X < 75)

Z-score for 75 = (75-mean)/standard deviation = (75-80)/8 = - 5/8 = -0.63

From standard normal tables, P(Z<-0.63) = 0.2643

P(X < 75) = P(Z<-0.63) = 0.2643

probability that a randomly chosen worker can repair equipment in less than an hour and 15 minutes =0.2643

b) Calculate the probability that a randomly chosen worker could take between 80 minutes and one hour ten minutes

= P(75 < X < 80) = P(X<80) - P(X<75)

Z-score 80 = (80-mean)/standard deviation = (80-80)=0

P(Z<0)=0.5

P(X<80) = P(Z<0)=0.5

From a) P(X<75) = 0.2643

P(75 < X < 80) = P(X<80) - P(X<75) = 0.5- 0.2643=0.2357

Calculate the probability that a randomly chosen worker could take between 80 minutes and one hour ten minutes=0.2357

c)

What is the probability of exactly 20 of the 100 workers, it takes between 80 minutes and one hour ten minutes

Number of workers : n=100

From b)

p: Probability that a randomly chosen worker could take between 80 minutes and one hour ten minutes = 0.2357

X: Number workers out of 100 would takes between 80 minutes and one hour ten minute

X follows Binomial distribution with n=100 and p = 0.2357

Probability mass function of X is given by

Probability that 'r' workers out of 100 would take between 80 minutes and one hour ten minute

probability of exactly 20 of the 100 workers, it takes between 80 minutes and one hour ten minutes= P(X=20)

probability of exactly 20 of the 100 workers, it takes between 80 minutes and one hour ten minutes= 0.068778347