Question

Columbia manufactures bowling balls with a mean weight of 14.7 pounds and a standard deviation of 2.5 pounds. A bowling ball is too heavy to use and is discarded if it weighs over 16 pounds. Assume that the weights of bowling balls manufactured by Columbia are normally distributed.

Round probabilities to four decimal places.

a) What is the probability that a randomly selected bowling ball is discarded due to being too heavy to use?

b) The lightest 5% of the bowling balls made are discarded due to the possibility of defects. A bowling ball is discarded for being too light if it weighs under what specific weight? (Round weight to two decimal places.)  pounds

c) What is the probability that a randomly selected bowling ball will be discarded for being either too heavy or too light?

Answer 1

mean weight = 14.7 pounds

standard deviation = 2.5 pounds.

A bowling ball is too heavy to use and is discarded if it weighs over 16 pounds.

a) What is the probability that a randomly selected bowling ball is discarded due to being too heavy to use?

X be the weight f ball in pounds

P[ X > 16 ] = P[ ( X - mean )/sd > ( 16 - mean )/sd ]

P[ X > 16 ] = P[ ( X - 14.7 )/2.5> ( 16 - 14.7 )/2.5 ]

P[ X > 16 ] = P[ Z > 0.52 ]

P[ X > 16 ] = 1 - P[ Z < 0.52 ]

P[ X > 16 ] = 1 - 0.6985

P[ X > 16 ] = 0.3015

Probability that a randomly selected bowling ball is discarded due to being too heavy to use = 0.3015

b) The lightest 5% of the bowling balls made are discarded due to the possibility of defects. A bowling ball is discarded for being too light if it weighs under what specific weight?

Let the weight be x_l

P[ X < x_l ] = 0.05

P[ ( X - mean )/sd < ( x_l - mean )/sd ] = 0.05

P[ ( X - 14.7 )/2.5 < ( x_l - 14.7 )/2.5 ] = 0.05

Also, P[ X < -1.645 ] = 0.05

Comparing

( x_l - 14.7 )/2.5 = 1.645

x_l - 14.7 = -1.645*2.5

x_l - 14.7 = -4.1125

x_l = 14.7 - 4.1125

x_l = 10.5875

A bowling ball is discarded for being too light if it weighs under 10.59 pounds

c) What is the probability that a randomly selected bowling ball will be discarded for being either too heavy or too light?

Probability that a randomly selected bowling ball will be discarded for being either too heavy or too light = Probability that a randomly selected bowling ball is discarded due to being too heavy to use + probability that a randomly selected bowling ball is discarded due to being too light to use.

Probability that a randomly selected bowling ball will be discarded for being either too heavy or too light = 0.3015 + 0.05

Probability that a randomly selected bowling ball will be discarded for being either too heavy or too light = 0.3515