Question

The amounts a soft drink machine is designed to dispense for each drink are normally​ distributed, with a mean of 11.9

fluid ounces and a standard deviation of 0.2 fluid ounce. A drink is randomly selected.

​(a) Find the probability that the drink is less than 11.7 fluid ounces.

​(b) Find the probability that the drink is between 11.6 and 11.7 fluid ounces.

​(c) Find the probability that the drink is more than 12.3 fluid ounces.

Can this be considered an unusual​ event? Explain your reasoning.

Answer 1

Part a)

P ( X < 11.7 )
Standardizing the value

Z = ( 11.7 - 11.9 ) / 0.2
Z = -1

P ( X < 11.7 ) = P ( Z < -1 )
P ( X < 11.7 ) = 0.1587

Part b)

P ( 11.6 < X < 11.7 )
Standardizing the value

Z = ( 11.6 - 11.9 ) / 0.2
Z = -1.5
Z = ( 11.7 - 11.9 ) / 0.2
Z = -1
P ( -1.5 < Z < -1 )
P ( 11.6 < X < 11.7 ) = P ( Z < -1 ) - P ( Z < -1.5 )
P ( 11.6 < X < 11.7 ) = 0.1587 - 0.0668
P ( 11.6 < X < 11.7 ) = 0.0918

Part c)

P ( X > 12.3 ) = 1 - P ( X < 12.3 )
Standardizing the value

Z = ( 12.3 - 11.9 ) / 0.2
Z = 2

P ( Z > 2 )
P ( X > 12.3 ) = 1 - P ( Z < 2 )
P ( X > 12.3 ) = 1 - 0.9772
P ( X > 12.3 ) = 0.0228

Part d)

Probability in Part c) is less than 5% i.e 0.05, hence it is unusual event.