Question

The weight of luggage has important implication for airplanes’ fuel consumption.

We Fly Everywhere (WFE) is a regional airline operating in NE USA. It has a small fleet of 12 small and mid-sized aircrafts. WFE operated on a narrow profit margin and has to take fuel consumption seriously and plan carefully.

From WFE’s point of view, the weight of a piece of luggage is unknown when a passenger books a flight. However, based on past experience it is known that the average piece of luggage weighs 20 lbs. and that the standard deviation is 5 lbs.

Questions

WFE randomly chooses one of its 50-seat airplanes and weighs all 100 pieces of luggage.

1. How is the sample mean distributed? What are its parameters and what are the parameters’ values?
2. What is the probability that the sample mean would be exactly 20 lbs.?
3. What is the probability that the sample mean would be less than 19 lbs.?
4. The maximum weight that the airplane can carry before WFE looses money due to high fuel costs is 2050 lbs. or 20.5 lbs. for the average piece of luggage. What are the chances the WFE would not lose money on this flight?

Answer 1

Population mean, = 20 lbs

Standard deviation, = 5 lbs

n = 100

The sampling distribution of the sample mean will be Normally distributed

It's parameters are Mean and Standard deviation

Mean,

Standard deviation,

Probability that the sample mean would be exactly 20 lbs ≈ 0 (Since probability at a point is 0 for any continuous distribution)

Probability that the sample mean would be less than 19 lbs = P{Z < (19 - 20)/0.5}

= P(Z < -2) = 0.0228

The chance that the WFE would not lose money on this flight = P(x < 20.5)

= P{Z < (20.5 - 20)/0.5}

= P(Z < 1) = 0.8413