# Math 243 Excel Assignment 2 (EA2)

Math 243 Excel Assignment 2 (EA2)

Random Variables

Name:

Practice questions are available on the last tab of this workbook.

Quest 1 Airlines commonly sell more tickets for a flight than there are seats.  They do this because not all  ticketed passengers actually show up for their flights.

Suppose an airline routinely sells 75 tickets for flights that have only 70 seats.  Let random variable X represent the number of passengers who actually show up at the airport for such a flight.  Suppose the airline has found, through prior experience, the probability distribution for X given below.

k P(X = k)

62 0.01

63 0.01

64 0.04

65 0.06

66 0.10

67 0.13

68 0.16

69 0.16

70 0.14

71 0.09

72 0.06

73 0.02

74 0.01

75 0.01

(A) P(X ≤ 70)

(B) Write a sentence or two explaining what your answer to part (a) means in the context of the problem.

(C) Fill out the table below to find E(X).

(D) Write a sentence or two explaining what E(X) means in the context of the problem.

Quest 2 This is the same scenario as the previous question.

Airlines commonly sell more tickets for a flight than there are seats.  They do this because not all  ticketed passengers actually show up for their flights.

Suppose an airline routinely sells 75 tickets for flights that have only 70 seats.  Let random variable X represent the number of passengers who actually show up at the airport for such a flight.  Suppose the airline has found, through prior experience, the probability distribution for X given below.

k P(X = k)

62 0.01

63 0.01

64 0.04

65 0.06

66 0.10

67 0.13

68 0.16

69 0.16

70 0.14

71 0.09

72 0.06

73 0.02

74 0.01

75 0.01

When more than 70 passengers show up for a flight, the airline asks for volunteers willing to take a later flight at no extra charge (this is called "bumping" a passenger).  These volunteers are each given a \$200 voucher they can use on another trip with the airline.

For example, if 72 passengers show up, it will cost the airline a total of \$400 to give vouchers to two passengers. If 70 or fewer passengers show up, there is no extra cost to the airline.

You may assume that the airline is always able to find enough volunteers to "bump" for each overbooked flight. You may also assume that there are always later flights with empty seats for these "bumped" volunteers.

(A) Find the distribution for random variable Y, the amount the airline has to pay in vouchers for a given flight of this type.

(B) For flights like this, what is the average amount, per flight, the airline pays for vouchers? Show your work and enter your answer in the specified cell below.

(C) This business practice is known as overbooking.  Write a sentence or two explaining why overbooking does or does not make financial sense for the airline.

Quest 3 This is the same scenario as the previous question.

Airlines commonly sell more tickets for a flight than there are seats.  They do this because not all  ticketed passengers actually show up for their flights.

Suppose an airline routinely sells 75 tickets for flights that have only 70 seats.  Let random variable X represent the number of passengers who actually show up at the airport for such a flight.  Suppose the airline has found, through prior experience, the probability distribution for X given below.

k P(X = k)

62 0.01

63 0.01

64 0.04

65 0.06

66 0.10

67 0.13

68 0.16

69 0.16

70 0.14

71 0.09

72 0.06

73 0.02

74 0.01

75 0.01

When more than 70 passengers show up for a flight, the airline asks for volunteers willing to take a later flight at no extra charge (this is called "bumping" a passenger).  These volunteers are each given a \$200 voucher they can use on another trip with the airline.

For example, if 72 passengers show up, it will cost the airline a total of \$400 to give vouchers to two passengers. If 70 or fewer passengers show up, there is no extra cost to the airline.

You may assume that the airline is always able to find enough volunteers to "bump" for each overbooked flight. You may also assume that there are always later flights with empty seats for these "bumped" volunteers.

(A) Fill out the table below to find the variance and standard deviation for X. Recall that μ is the same thing as expected value. Be sure to use the value you computed in question 1.

(B) Plot a histogram for the probability distribution of X.  Make sure your histogram has an appropriate title,  a gap width of zero, and borders around each bar.  Put your graph in the empty space to the right.

(C) If you have plotted the histogram correctly, you should see that the distribution is approximately bell-shaped. Let's see if this distribution follows the empirical rule.

(D) Are the probabilities for random variable X reasonably close to the empirical rule values? Answer yes/no, and write a sentence or two explaining your answer.  Be sure your answer is written in the context of the problem (seats on an overbooked flight).

Suppose a student guesses on all five questions of a multiple choice test, where each question has four choices.

Let random variable X represent the number of questions answered correctly.  The pdf for X is given below.

k 0 1

P(X = k) 0.237 0.395

For practice, find the following.  The correct answers are given on the right-hand part of the worksheet.

Find the following:

Find E(X)

Interpret E(X)

Find Var(X)

Find σX

Interpret σX

Graph X

Does X follow the empirical rule?