# Samples

A sample of 12 homes sold last week in St. Paul, Minnesota, is selected. Can we conclude that as the size of the home (reported below in thousands of square feet) increases, the selling price (reported in $ thousands) also increases?

Home Size Selling Home Size Selling

(thousands Price (thousands of Price

of square feet) ($ thousands) square feet) ($ thousands)

1.4 100 1.3 110

1.3 110 .08 85

1.2 105 1.2 105

1.1 120 0.9 75

1.4 80 1.1 70

1.0 105 1.1 95

Following is a regression equation.

Y = 17.08 + 0.16X

This information is also available: S = 4.05, X = 210, X = 9,850 and n = 5.

a. Estimate the value of Y when X = 50.

b. Develop a 95 percent prediction interval for an individual value of Y for X = 50.

The proprietor of the newly built Ski and Swim Lodge has been considering purchasing or leasing several snowmobiles for the use of guests. The owner found that other financial obligations made it impossible to purchase the machines. Snowmobiles Incorporated (SI) will lease a machine for $20 a week, including any needed maintenance. According to SI, the usual rental charge to the guests of the lodge is $25 a week. Gasoline and oil are extra. Snowmobiles Incorporated only leases a machine for the full season. The proprietor of Ski and Swim, knowing that leasing an excessive number of snowmobiles might cause a net loss for the lodge, investigated the records of other resort owners. The combined experience at several other lodges was found to be:

Number of

Snowmobiles

Demanded Number of

By Guests Weeks

7 10

8 25

9 45

10 20

a. Design a payoff table.

b. Compute the expected profits for leasing 7,8,9, and 10 snowmobiles based on the cost of leasing of $20, the rental charge of $25, and the experience of other lodges.

c. Which alternative is the most profitable?

d. Design an opportunity loss table.

e. Find the expected opportunity losses for leasing 7,8,9 and 10 snowmobiles.

f. Which act would give the least expected opportunity loss?

Tim Waltzer owns and operates Waltzer's Wrecks, a discount car rental agency near the Cleveland Hopkins International Airport. He rents a wreck for $20 a day. He has an arrangement with Landrum Leasing to purchase used cars at $6000 each. His cars receive only needed maintenance and, as a result, are worth only $2000 at the end of the year of operation. Tim has decided to sell all his wrecks every year and purchase a complete set of used cars from Landrum Leasing.

His clerk-accountant provided him with a probability distribution with respect to the number of cars rented per day.

Numbers of Cars Rented per Day

20 21 22 23

Probability .1 .2 .5 .2

Tim is an avid golfer and tennis player. He is either on the golf course on weekends or playing tennis indoors. Thus, his car rental agency is only open weekdays. Also, he closes for two weeks during the summer and goes on a golfing tour.

The clerk-accountant estimated that it cost $1.50 per car rental for minimal maintenance and cleaning.

a. How many cars should he purchase to maximize profit?

b. What is the expected value of perfect information?

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The solution discusses A sample of 12 homes sold last week in St. Paul, Minnesota, is selected. Can we conclude that as the size of the home (reported below in thousands of square feet) increases, the selling price (reported in $ thousands) also increases?

Home Size Selling Home Size Selling

(thousands Price (thousands of Price

of square feet) ($ thousands) square feet) ($ thousands)

1.4 100 1.3 110

1.3 110 .08 85

1.2 105 1.2 105

1.1 120 0.9 75

1.4 80 1.1 70

1.0 105 1.1 95

Following is a regression equation.

Y = 17.08 + 0.16X

This information is also available: S = 4.05, X = 210, X = 9,850 and n = 5.

a. Estimate the value of Y when X = 50.

b. Develop a 95 percent prediction interval for an individual value of Y for X = 50.

The proprietor of the newly built Ski and Swim Lodge has been considering purchasing or leasing several snowmobiles for the use of guests. The owner found that other financial obligations made it impossible to purchase the machines. Snowmobiles Incorporated (SI) will lease a machine for $20 a week, including any needed maintenance. According to SI, the usual rental charge to the guests of the lodge is $25 a week. Gasoline and oil are extra. Snowmobiles Incorporated only leases a machine for the full season. The proprietor of Ski and Swim, knowing that leasing an excessive number of snowmobiles might cause a net loss for the lodge, investigated the records of other resort owners. The combined experience at several other lodges was found to be:

Number of

Snowmobiles

Demanded Number of

By Guests Weeks

7 10

8 25

9 45

10 20

a. Design a payoff table.

b. Compute the expected profits for leasing 7,8,9, and 10 snowmobiles based on the cost of leasing of $20, the rental charge of $25, and the experience of other lodges.

c. Which alternative is the most profitable?

d. Design an opportunity loss table.

e. Find the expected opportunity losses for leasing 7,8,9 and 10 snowmobiles.

f. Which act would give the least expected opportunity loss?

Tim Waltzer owns and operates Waltzer's Wrecks, a discount car rental agency near the Cleveland Hopkins International Airport. He rents a wreck for $20 a day. He has an arrangement with Landrum Leasing to purchase used cars at $6000 each. His cars receive only needed maintenance and, as a result, are worth only $2000 at the end of the year of operation. Tim has decided to sell all his wrecks every year and purchase a complete set of used cars from Landrum Leasing.

His clerk-accountant provided him with a probability distribution with respect to the number of cars rented per day.

Numbers of Cars Rented per Day

20 21 22 23

Probability .1 .2 .5 .2

Tim is an avid golfer and tennis player. He is either on the golf course on weekends or playing tennis indoors. Thus, his car rental agency is only open weekdays. Also, he closes for two weeks during the summer and goes on a golfing tour.

The clerk-accountant estimated that it cost $1.50 per car rental for minimal maintenance and cleaning.

a. How many cars should he purchase to maximize profit?

b. What is the expected value of perfect information?

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Sampling Distribution, Mean and Standard Deviation

See attachment for better symbol representation.

1) A manufacturer of paper used for packaging requires a minimum strength of 20 pounds per square inch. To check on the quality of the paper, a random sample of 10 pieces of paper is selected each hour from the previous hour's production and a strength measurement is recorded for each. The standard deviation σ of the strength measurements, computed by pooling the sum of squares of deviations of many samples, is know to equal 2 pounds per square inch, and the strength measurements are normally distributed.

a) What is the approximate sampling distribution of the sample mean of n = 10 test pieces of paper?

b) If the mean of the population of strength measurements is 21 pounds per square inch, what is the approximate probability that, for a random sample of n = 10 test pieces of paper, ¯x < 20?

c) What value would you select for the mean paper strength μ in order that P (¯x < 20) be equal to .001?

2) Suppose a random sample of n = 25 observations is selected from a population that is normally distributed, with mean equal to 106 and standard deviation equal to 12?

a) Give the mean and standard deviation of the sampling distribution of the sample mean ¯x.

b) Find the probability that ¯x exceeds 110

c) Find the probability that the sample mean deviates from the population mean μ = 106 by no more than 4.