Question

John knows that monthly demand for his product follows a normal distribution with a mean of 2,500 units and a standard deviation of 425 units. Given this, please provide the following answers for John.

a. What is the probability that in a given month demand is less than 3,000 units?

b. What is the probability that in a given month demand is greater than 2,200 units?

c. What is the probability that in a given month demand is between 2,200 and 3,000 units?

d. What is the probability that demand will exceed 5,000 units next month?

e. If John wants to make sure that he meets monthly demand with production output at least 95% of the time. What is the minimum he should produce each month?

Answer 1

Part a)

P ( X < 3000 )
Standardizing the value

Z = ( 3000 - 2500 ) / 425
Z = 1.18

P ( X < 3000 ) = P ( Z < 1.18 )
P ( X < 3000 ) = 0.881

Part b)

P ( X > 2200 ) = 1 - P ( X < 2200 )
Standardizing the value

Z = ( 2200 - 2500 ) / 425
Z = -0.71

P ( Z > -0.71 )
P ( X > 2200 ) = 1 - P ( Z < -0.71 )
P ( X > 2200 ) = 1 - 0.2389
P ( X > 2200 ) = 0.7611

Part c)

P ( 2200 < X < 3000 )
Standardizing the value

Z = ( 2200 - 2500 ) / 425
Z = -0.71
Z = ( 3000 - 2500 ) / 425
Z = 1.18
P ( -0.71 < Z < 1.18 )
P ( 2200 < X < 3000 ) = P ( Z < 1.18 ) - P ( Z < -0.71 )
P ( 2200 < X < 3000 ) = 0.8803 - 0.2401
P ( 2200 < X < 3000 ) = 0.6402

Part d)

P ( X > 5000 ) = 1 - P ( X < 5000 )
Standardizing the value

Z = ( 5000 - 2500 ) / 425
Z = 5.88

P ( Z > 5.88 )
P ( X > 5000 ) = 1 - P ( Z < 5.88 )
P ( X > 5000 ) = 1 - 1
P ( X > 5000 ) = 0

Part e)

P ( Z > ? ) = 0.95%

P ( Z > ? ) = 1 - 0.95 = 0.05

Looking for the probability 0.05 in standard normal table to find critical value Z

Z = -1.64