Question

A car manufacturer has determined it takes an average time of 54 minutes to produce a car. The population standard deviation is assumed to be 4 minutes. The company pays a bonus to the workers for every car produced in 46 minutes or less. Assuming that the production time is normally distributed, answer the following questions. Let X = production time of a randomly selected car.

Round all probabilities to four decimal places and times to two decimal places.

a) What is the probability that the workers will receive the bonus?

b) Suppose on a certain day we sampled 6 cars that were produced and looked at their average production time. What is the probability that the average production time was more than one hour?

c) Between what two times do the middle 70% of the average production times fall? and

d) Of these 6 sampled cars, suppose we look at each one to see whether it was completed within the employee bonus time frame. What is the probability that between 2 and 4 cars (inclusive) were produced within the bonus time frame?

e) What is the probability that exactly 3 cars were produced within the bonus time frame?

Answer 1

Answer:

a)

To determine the probability that the workers will receive the bonus i.e.,

= P(X<46)

= P(Z<(46-54)/4)

= P(Z<-2)

Required probability = 0.0228

b)

Given,

n = 6

mean = 54

std error of mean = std deviation/sqrt(n)

= 4/sqrt(6)

= 1.633

So the probability that the average production time which was more than one hour

= P(X>60)

= P(Z>(60-54)/1.633)

= P(Z>3.67)

Required probability = 0.0001

c)

for the middle 70% values ;

crtiical value

z = +/- 1.04

So the middle 70% vlaues = mean +/- z * std deviation

= 54 +/- 1.04 * 1.633

= 52.30 to 55.70

So between 52.30 and 55.70 average production fall

d)

To determine the required probability

Assume the binomial distribution

p = 0.0228

n = 6

Now the probability that between 2 and 4 cars were produced within the bonus time

=

On solving then we get

Required probability = 0.0073

e)

To determine the probability that exactly 3 cars were produced within bonus time frame

=   

On solving then we get our required probability

= 0.0002