Question

A second hand car dealer has 7 cars for sale. She decides to investigate the link between the age of the cars, x years, and the mileage, y thousand miles. The date collected from the cars is shown in the table below.

Age, x Year

2

3

7

6

4

5

8

Mileage, y thousand

20

18

15

24

29

21

20

Use your line to find the mileage predicted by the regression line for a 20 year old car.

a.

243

b.

21

c.

15

d.

234

A second hand car dealer has 7 cars for sale. She decides to investigate the link between the age of the cars, x years, and the mileage, y thousand miles. The date collected from the cars is shown in the table below.

Age, x Year

2

3

7

6

4

5

8

Mileage, y thousand

20

18

15

24

29

21

20

Find the least square regression line in the form y = a + bx.

a.

Y= 23- 0.4 X

b.

Y= 23 + 4 X

c.

Y= 10 + 53 X

d.

Y= 43 + 10 X

Each coffee table produced by Robert West Designers nets the firm a profit of $9. Each bookcase yields a $12 profit. West’s firm is small and its resources limited. During any given production period, 10 gallons of varnish and 12 lengths of high-quality redwood are available. Each coffee table requires approximately 1 gallon of varnish and 1 length of redwood. Each bookcase takes 1 gallon of varnish and 2 lengths of wood.

Formulate West’s production-mix decision as a linear programming problem, and solve. How many tables and bookcases should be produced each week? What will the maximum profit be?

Use:

x = number of coffee tables to be produced
y = number of bookcases to be produced

Which objective function best represents the problem?

a.

P= 9 X + 12 Y

b.

P= 10 X + 12 Y

c.

P= X + Y

d.

P= X + 2 Y

Each coffee table produced by Robert West Designers nets the firm a profit of $9. Each bookcase yields a $12 profit. West’s firm is small and its resources limited. During any given production period, 10 gallons of varnish and 12 lengths of high-quality redwood are available. Each coffee table requires approximately 1 gallon of varnish and 1 length of redwood. Each bookcase takes 1 gallon of varnish and 2 lengths of wood.

Formulate West’s production-mix decision as a linear programming problem, and solve. How many tables and bookcases should be produced each week? What will the maximum profit be?

Use:

x = number of coffee tables to be produced
y = number of bookcases to be produced

For the problem above, what is the optimal solution?

a.

96

b.

72

c.

90

d.

98

Answer 1

First, let's find the equation of the regression line using the given data:

Using a calculator or spreadsheet, we can find that the slope of the regression line is -1.35 and the y-intercept is 26.5.

Therefore, the equation of the regression line is:
y = -1.35x + 26.5

To find the mileage predicted by the regression line for a 20 year old car, we can substitute x = 20 into the equation:
y = -1.35(20) + 26.5 = 0.5

Therefore, the predicted mileage for a 20 year old car is 0.5 thousand miles, or 500 miles.

Answer: b. 21

To find the least square regression line in the form y = a + bx, we need to use the formula:
b = Σ[(xi - x)(yi - y)] / Σ(xi - x)^2
a = y - bx

where x and y are the sample means, xi and yi are the individual data points, and Σ is the sum of the values.

Using the given data, we can calculate:
x = (2+3+7+6+4+5+8) / 7 = 5
y = (20+18+15+24+29+21+20) / 7 = 21.43

Σ(xi - x)^2 = (2-5)^2 + (3-5)^2 + (7-5)^2 + (6-5)^2 + (4-5)^2 + (5-5)^2 + (8-5)^2 = 56
Σ[(xi - x)(yi - y)] = (2-5)(20-21.43) + (3-5)(18-21.43) + (7-5)(15-21.43) + (6-5)(24-21.43) + (4-5)(29-21.43) + (5-5)(21-21.43) + (8-5)(20-21.43) = -121.43

Therefore, b = -121.43 / 56 = -2.17
a = 21.43 - (-2.17)(5) = 32.28

Therefore, the equation of the least square regression line is:
y = 32.28 - 2.17x