Question

The weight of packages carried by a certain post office is a normal random variable with a mean of 10 pounds and a standard deviation of 2 pounds. The post office wants to apply an additional charge to packages containing m160s of a certain weight, but they want to set the policy in such a way that the surcharge only applies to 1% of customers.
1) What is the weight at which you must begin to apply additional charge?
2) If they decide to apply the additional charge to packages weighing more than 12 pounds, what proportion of customers will be affected?
3) If you decide to apply the charge to packages of 12 pounds or more, and during a randomly selected day 32 clients arrive with packages. What is the probability that at least 10 of them will have to pay the charge?
4) If the additional charge is $ 5.00 and on an average random day 378 packages are received, what is the expected value of the amount of money that will be collected for the additional charge each day?

Answer 1

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

Therefore the cutoff weight value for surcharge is computed here as
= Mean + 2.326*Std Dev

= 10 + 2.326*2 = 14.652
therefore 14.652 pounds is the required cutoff value here.

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

Converting it to a standard normal variable:
P(Z > (12 - 10)/2)
P(Z > 1)

Getting it from the standard normal tables, we have:
P(Z > 1) = 0.1587

Therefore 0.1587 is the required proportion here.

3) The number with surcharge out of the sample size of 32 is modelled here as

Therefore the probability that at least 10 of them will have to pay the charge is computed here as:
P(X >= 10) = 1 - P(X <= 9)

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.

4) The expetced number of total additional charge each day is computed here as:
= 5np

= 5*378*0.1587

= $299.943

therefore $299.943 is the required additional charge expected per day here.