Probability, Linear Programming and Statistics
Show your work.
1. (3 points) Find the equation of the line shown:
2. (6 points) A bank loaned $15,000, some at an annual rate of 16% and some at an annual rate of 10%. If the income from these loans was $1800, how much was loaned at 10%?
3. (3 point) Write the augmented matrix of the system:
2x1 -3x2 + x3 = 0 x1 - x2 - 2x3 = -2 -5x1 + x2 = -4
4. (3 point) Perform the row operation R2 = (-2)r1 + r2 on the matrix
5. (3 point) Indicate whether the reduced row-echelon form of each augmented matrix has one solution, no solution, or infinitely solutions.
a. b. c.
6. (4 points) Find: .
7. (4 points) Find: .
8. (6 Points) Find the inverse of: .
9. (8 points) Maximize P = x1 + 2x2 + 3x3 using the simplex method.
subject to the constraints
2x1 + x2 + x3 < 25
2x1 + 3x2 + 3x3 < 30
x1 > 0, x2 > 0, x3 > 0,
10. (8 points) A company makes three products, A, B, and C. There are 500 pounds of raw material available. Each unit of product A requires 2 pounds of raw material, each unit of product B requires 2 pounds of raw material, and each unit of product C requires 3 pounds. The assembly line has 1,000 hours of operation available. Each unit of product A requires 4 hours, while each unit of products B and C requires 5 hours. The company realizes a profit of $500 for each unit of product A, $600 for each unit of product B, and $1,000 for each unit of product C. Formulate (but don't solve) a linear program to determine how many units of each of the three products the company should make to maximize profits.
11. (3 point) What is the ending balance from an initial deposit of $4,250 at 12% compounded quarterly for 6 years?
12. (3 points) Find the present value of $5,000 in 5 years at 10% compounded annually.
13. (4 points) Find the value of an annuity in which $1,100 is deposited at the end of each year for 5 years, at an interest rate of 11.5% compounded annually.
14. (4 points) Determine the amount of each payment to be made to a sinking fund in order to pay off a $120,000 loan in 8 1/2 years when the funds earn interest at a rate of 10% compounded semiannually.
15. (4 points) Find the present value of an ordinary annuity with annual payments of $1,000, for 6 years, at 10% interest compounded annually.
16. (2 points) In a marketing survey, consumers are asked to give their first three choices, of 9 different drinks. In how many different ways can they indicate their choices?
17. (6 points) A class consists of 15 students. The instructor wants to pick a group of 4 to work on a special project.
a. How many different groups of 4 can he choose?
b. If the class consists of 10 girls and 5 boys, how many different groups
of 4 are made up of 2 boys and 2 girls?
18. (4 points) How many license plates can a state have if each license plate has 2 letters followed by 4 digits, if the first digit cannot be zero?
19. (6 points) Two cards are drawn at random from an ordinary deck of playing cards. The first is not replaced before the second is drawn. What is the probability that:
a. Both cards are aces?
b. At least one card is black?
20. (16 points) Using the following sample: 28, 30, 24, 30, 32, 40, 22, 25, 26, 34
a. Find the mean.
b. Find the median.
c. Find the mode.
d. Find the standard deviation.
e. Find the z-score for 30. (Assuming a normal distribution)
f. Find the range.