Question

Beer bottles are filled so that they contain an average of 480 ml of beer in each bottle. Suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml. [You may find it useful to reference the z table.]

a. What is the probability that a randomly selected bottle will have less than 474 ml of beer? (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

b. What is the probability that a randomly selected 6-pack of beer will have a mean amount less than 474 ml? (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What is the probability that a randomly selected 12-pack of beer will have a mean amount less than 474 ml? (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Answer 1

a) P( X< 474) = P( Z < [(474- 480)/8])

P(Z < -0.75) = 0.22663

About 22.663% of the bottles will have less than 474 ml of beer.

b) To calculate the probability that the mean of a sample of 6 would be 1 or more standard deviations below the population mean, we use the following formula to calculate the z-value or standard normal value:

Z = (X - 480)/ (8/6^0.5)

as n= 6

P( Z < ( 474- 480)/(8/√6) )

= P(Z< -1.837)

=P( Z< -1.84) = .03288

so , About 32.88% of the 6 pack bottles will have less than 474 ml of beer.

c) n = 12

Z = (X - 480)/ (8/12^0.5)

as n= 12

P( Z < ( 474- 480)/(8/√12) )

= P(Z< -2.598)

=P( Z< -2.60) = .00466

so , About 0.466%% of the 12 pack bottles will have less than 474 ml of beer.