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 0.04 liter. Suppose you select a random sample of 25 bottles.

a. What is the probability that the sample mean will be between 1.99 and 2.0 liters ?

b. What is the probability that the sample mean will be below 1.98 liters ?

c. What is the probability that the sample mean will be 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 values (symmetrically distributed around the mean)?

a. The probability is ____ (Round to three decimal places as needed.)

b.The probability is ____. (Round to three decimal places as needed.)

c. The probability is____. (Round to three decimal places as needed.)

d. There is a 99 % probability that the sample mean amount of soft drink will be at least ____liter(s). (Round to three decimal places as needed.)

e. There is a 99 % probability that the sample mean amount of soft drink will be between ____liter(s) and nothing liter(s). (Round to three decimal places as needed. Use ascending order.)

Answer 1

Part a)

Standardizing the value

Part b)

Standardizing the value

Part c)

Part d)

P ( Z > ? ) = 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

Z = -2.33

Part e)

P ( z < Z < z ) = 0.99

Symmetric around the mean means that 0.5 area under the curve is above and below the mean

0.99 / 2 = 0.495

Area above the mean = 0.5 + 0.495 = 0.995

Area below the mean = 0.5 - 0.495 = 0.005

Finding the critical value Z for the probability 0.995 and 0.005 is Z = 2.58 and Z = -2.58

P ( 1.979 < X < 2.021 ) = 99%