Question

The fill amount of bottles of a soft drink is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be:

A.) Between 1.99 and 2.0 liters

B.) Below 1.98 liters

C.) Greater than 2.01 liters

D.) The probability is 99% that the sample mean amount of soft drink will be at least how much?

E.) The probability is 99% that the sample mean amount of soft drink will be between which two amounts?

Answer 1

Part a)

P ( 1.99 < X < 2 )

Standardizing the value

Z = -1

Z = 0

P ( -1 < Z < 0 )

P ( 1.99 < X < 2 ) = P ( Z < 0 ) - P ( Z < -1 )

P ( 1.99 < X < 2 ) = 0.5 - 0.1587

P ( 1.99 < X < 2 ) = 0.3413

Part b)

P ( X < 1.98 )

Standardizing the value

Z = -2

P ( X < 1.98 ) = P ( Z < -2 )

P ( X < 1.98 ) = 0.0228

Part c)

P ( X > 2.01 ) = 1 - P ( X < 2.01 )

Standardizing the value

Z = 1

P ( Z > 1 )

P ( X > 2.01 ) = 1 - P ( Z < 1 )

P ( X > 2.01 ) = 1 - 0.8413

P ( X > 2.01 ) = 0.1587

Part d)

P ( Z > ? ) = 99% = 0.99

P ( Z > ? ) = 1 - P ( Z < ? ) = 1 - 0.99 = 0.01

Looking for the probability 0.01 in standard normal table to find the critical value z = -2.33

Part e) P ( a < Z < b ) = 99% = 0.99 Area above the mean = 0.5 + ( 0.99 / 2 ) = 0.5 + 0.495 = 0.995 Looking for the probability 0.995 in standard normal table to find the critical value Z = 2.58 P ( 1.97 < X < 2.03 ) = 99%