Question

A manufacturing company measures the weight of boxes before shipping them to the customers. The box weights have a population mean and standard deviation of 86 lbs. and 16 lbs. respectively.

a. Based on a sample size of 64 boxes, the probability that the average weight of the boxes will exceed 90.3 lbs. is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

b. Based on a sample size of 64 boxes, the probability that the average weight of the boxes will be between 82.3 and 84.5 lbs. is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

c. Based on a sample size of 64 boxes, the probability that the average weight of the boxes will be less than 87.7 lbs. is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

d. If random samples of 64 boxes are repeatedly drawn from this population and the mean number of ounces in each sample is recorded, 82.9% of the sample means should be above  pounds. Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 2 decimal places, using conventional rounding rules.

Answer 1

a) P( > 90.3)

= P(( - )/() > (90.3 - )/())

= P(Z > (90.3 - 86)/(16/))

= P(Z > 2.15)

= 1 - P(Z < 2.15)

= 1 - 0.9842

= 0.0158

b) P(82.3 < < 84.5)

= P((82.3 - )/() < ( - )/() < (84.5 - )/())

= P((82.3 - 86)/(16/) < Z < (84.5 - 86)/(16/))

= P(-1.85 < Z < -0.75)

= P(Z < -0.75) - P(Z < -1.85)

= 0.2266 - 0.0322

= 0.1944

c) P( < 87.7)

= P(( - )/() < (87.7 - )/())

= P(Z < (87.7 - 86)/(16/))

= P(Z < 0.85)

= 0.8023

d) P( > x) = 0.829

or, P(( - )/() > (x - )/()) = 0.829

or, P(Z > (x - 86)/(16/)) = 0.829

or, P(Z < (x - 86)/(16/)) = 0.171

or, (x - 86)/(16/) = -0.95

or, x = -0.95 * (16/) + 86

or, x = 84.10