bus

Business Statistics

Homework 2 (100 pts)

Remember, all work should be your own. Please submit your homework online, by the

beginning of class on 4/14. Late submission will be penalized.

Problem 1 (10 points)

In a recent survey about appliance ownership, 58.3% of the respondents indicated that they own

Maytag appliances, while 23.9% indicated they own both Maytag and GE appliances and 70.7% said

they own at least one of the two appliances.

Define the events as

M=Owning a Maytag appliance

G=Owning a GE appliance

a. What is the probability that a respondent owns a GE appliance?

b. Given that a respondent owns a Maytag appliance, what is the probability that the respondent

also owns a GE appliance?

c. Are events "M" and "G" mutually exclusive? Why or why not? Explain, using probabilities.

d. Are the two events "M" and "G" independent? Explain, using probabilities.

Problem 2 (14 points)

A government agency has 6,000 employees. The employees were asked whether they preferred a

four-day work week (10 hours per day), a five-day work week (8 hours per day), or flexible hours.

You are given information on the employees' responses broken down by sex.

Male Female Total

Four days 300 600 900

Five days 1,200 1,500 2,700

Flexible 300 2,100 2,400

Total 1,800 4,200 6,000

a. What is the probability that a randomly selected employee is a man and is in favor of a four-day

work week?

b. What is the probability that a randomly selected employee is female?

c. A randomly selected employee turns out to be female. Compute the probability that she is in

favor of flexible hours.

d. What percentage of employees is in favor of a five-day work week?

e. Given that a person is in favor of flexible time, what is the probability that the person is female?

f. What percentage of employees is male and in favor of a five-day work week?

Problem 3 (8 points)

A machine is used in a production process. From past data, it is known that 97% of the time the

machine is set up correctly. Furthermore, it is known that if the machine is set up correctly, it

produces 95% acceptable (non-defective) items. However, when it is set up incorrectly, it produces

only 40% acceptable items.

a. An item from the production line is selected. What is the probability that the selected item is

non-defective?

b. Given that the selected item is non-defective, what is the probability that the machine is set up

correctly?

Problem 4 (8 points)

Records of a company show that 20% of the employees have only a high school diploma; 70% have

bachelor degrees; and 10% have graduate degrees. Of those with only a high school diploma, 10%

hold management positions; whereas, of those having bachelor degrees, 40% hold management

positions. Finally, 80% of the employees who have graduate degrees hold management positions.

a. What percentage of employees holds management positions?

b. Given that a person holds a management position, what is the probability that she/he has a

graduate degree?

Problem 5 (10 points)

One of your Class friends, claiming that he is a real estate specialist, provides a report with

information for the real estate investment decision, forming conditional probabilities:

Possible outcomes

g=good economic conditions

p=poor economic conditions

P=positive economic report

N=negative economic report

P(P | g)=0.8 P(N | g)=0.2 P(P | p)=0.1 P(N | p)=0.9

You also know the prior probabilities P(g)=0.6 and P(p)=0.4

What is the posterior probability P(g┃P)=?

Problem 6 (6 points)

A company sells its products to wholesalers in batches of 1,000 units only. The daily demand for its

product and the respective probabilities are given below.

Demand (Units) Probability

0 0.2

1000 0.2

2000 0.3

3000 0.2

4000 0.1

a. Determine the expected daily demand.

b. Assume that the company sells its product at $3.75 per unit. What is the expected daily revenue?

Problem 8 (14 points)

The size of the price discount offered by an industrial marketing manager varies according to

order size. During the last quarter, the manager offered price discounts of 20% to one half of his

customers and 30% to one-quarter of his customers, and no discount to the rest. Assume no

customer received more than one discount. He estimates that the probability of receiving a

reorder this quarter is 0.90 if the customer received a 20% discount, 0.92 if the customer received

a 30% discount, and 0.70 if the customer received no discount.

a) If a customer is selected at random and is observed not to have reordered, what is the

probability that this customer was offered no discount last quarter? (8 points)

b) If a customer is selected at random and is observed to have reordered, what is the

probability that this customer was offered a 30\% discount last quarter? (6 points)

Problem 9 (12 points)

The probability function for the number of insurance policies John will sell to a customer is

given by

f(X) = .5 - (X/6) for X = 0, 1, or 2

a) Is this a valid probability function? Explain your answer.

b) What is the probability that John will sell exactly 2 policies to a customer?

c) What is the probability that John will sell at least 2 policies to a customer?

d) What is the expected number of policies John will sell?

e) What is the variance of the number of policies John will sell?

Problem 10 (10 points)

A manufacturing company has 5 identical machines that produce nails. The probability that a

machine will break down on any given day is .1. Define a random variable X to be the number

of machines that will break down in a day.

a. What is the appropriate probability distribution for X? Explain how X satisfies the

properties of the distribution.

b. Compute the probability that 4 machines will break down.

c. Compute the probability that at least 4 machines will break down.

d. What is the expected number of machines that will break down in a day?

e. What is the variance of the number of machines that will break down in a day?

Problem 11 (8 points)

On the average, 6.7 cars arrive at the drive-up window of a bank every hour. Define the random

variable X to be the number of cars arriving in any hour.

a. What is the appropriate probability distribution for X? Explain how X satisfies the

properties of the distribution.

b. Compute the probability that exactly 5 cars will arrive in the next hour.

c. Compute the probability that no more than 5 cars will arrive in the next hour.