Question

The weight of the packages carried by a certain post office is a Normal random variable with a mean of 10 lbs and a standard deviation of 2 Ib. The post office wants to apply an additional charge (S) to packages that contain more than a certain weight, but they want to establish the policy so that the surcharge only applies to 1% of customers.

a) What is the weight to which the additional charge should begin to be applied?

b) If you decide to apply the additional charge to packages weighing more than 12 Ibs, what proportion of the clients will be affected?

c) If it is decided to apply the charge to packages of 12-lbs or more. and during a randomly selected day 32 clients arrive with packages, what is the probability that at least 10 of them have to pay the charge?

d) If the additional charge is $ 5.00 and 378 packages are received on average on a random day, what is the expected value of the amount of money that will be collected for the additional charge each day?

Answer 1

a) From standard normal tables, we have:
P(Z > 2.326) = 0.01

Therefore, the weight cutoff is computed here as:
= Mean + 2.326*Std Dev

= 10 + 2.326*2 = 14.652 lbs

Therefore 14.652 lbs is the required cutoff weight here.

b) The proportion here is computed as:
P(X > 12)

Converting it to a standard normal variable, we have here:
P(Z > (12 - 10) /2)

= P(Z > 1)

Getting it from the standard normal tables, we have here:
P(Z > 1) = 0.1587
Therefore 0.1587 is the required proportion here.

c) The number of participants out of 32 that will have to pay extra could be modelled here as:

Probability that at least 10 have to pay the surcharge is computed here as:
P(X >= 10)

This is computed in EXCEL as:
=1-binom.dist(9,32,0.1587,TRUE)

0.0226 is the output here.

Therefore 0.0226 is the required probability here.

d) The expected money collected in surcharges is computed here as:

= 5*Expected number of packages with surcharge amount

= 5*np

= 5*32*0.1587

= 25.392

Therefore $25.392 is the required expected amount here.