## Math 160: Study Guide - Final Exam

1. Compound interest problem. Use the "finance" program.
2. Present value problem. Use the "finance" program.
3. Retirement problem. Figure out what the need to retire and what it will take each month to save that. Figure out how long the retirement will last if extra is paid. Find the remaining balance after a certain period of time. Use the "finance" program.
4. Matrix multiplication problem. Find the product of two matrices, interpret the product by labeling the rows and columns. Look at the boat problem out of section 4.4.
5. Solve a 3x3 system of linear equations. Either use Gauss-Jordan elimination (pivoting) or use the inverse of a matrix.
6. Solve the matrix equation for X. Two parts.
7. Leontief input output problem. Use the "leontief" program.
8. List the requirements for a linear programming problem to be in standard form.
9. The initial tableau from a non-standard maximization problem is given. Write the maximization problem.
10. Maximize a standard linear programming problem. Write the initial tableau and final tableau, and give the values of the objective function and decision variables. Use the "simplex" program.
11. Given P(A), P(B) and one more probability, complete a probability distribution and then find several probabilities from the probability distribution.
12. Find probabilities. Part of it can be done using the hypergeometric distribution, but you will also need to use the multiplication rule for independent events.
13. Find the mean, median, and sample standard deviation for a set of data.
14. Binomial probabilities. Use the "binom" program.
15. Solve a game matrix, giving the optimal row and column strategies, and the value of the game. Use the "game" program.
16. Markov chain problem. Write the initial state matrix, the transition matrix. Find the first state matrix, and the steady state matrix.
17. Linear programming problem. A tableau is given. Label the columns as appropriate using x's for the decision variables, s's for the slack variables, z for the objective function, and rhs for the right hand side. Identify the value of the variables. Choose a possible objective function (multiple choice). Identify which variables are basic and which are non-basic. Circle the pivot element. Identify which variable is entering and which is exiting.
18. Decision Theory. Most of the payoff table has been created for you, but you need to find two entries in the table. Create the opportunistic loss table. Then give the value and optimal action under each criteria.
19. Probability problem. Two bags with different types of coins in them. Draw a coin out of bag 1 and place it into bag 2 and then find some probabilities.
20. The payoffs for a game are given. Find the probabilities of each payoff and then find the expected value. Identify whether the game is fair or not. You may use the "pdist" program, although it's probably easier to do by hand.
21. The final tableau from a zero-sum, two-player game is given. Give the optimal row and column strategies and the value of the game.
22. Absorbing Markov chain problem. Write the initial state matrix and the transition matrix. Then find the expected number of transient states before leaving the matrix.

### Notes:

• The test is open notebook. Make sure that your notes are complete in the sections covered on the exam.
• Problems 1-16 are to be worked individually. Problems 17-22 may be worked with a partner. You must turn in problems 1-16 before getting into a group.
• The plan is to spend the first day on the first part and the second day on the second part, however, if you finish early on the first day, you may begin the second part of the test.
• When you get into partners, you will lose the use of your notebooks. For this reason, get as much of the information transferred from your notes onto your test as necessary before you get into partners. You may wish to start the second part alone with your notes, answer as much as you can, and then get into the groups when you have exhausted your notes.
• The exam is laid out in chapter order. Having your notes in order will help.
• The individual part of the exam is 125 points, the group part is 75 points.

Point break down by problem

 # 1 2 3 4 5 6 7 8 9 10 11 Pts 5 5 10 6 6 6 6 6 6 9 14 12 13 14 15 16 17 18 19 20 21 22 Total 6 9 9 10 12 14 17 12 12 10 10 200

Point break down by chapter

 Chapter 3 4 5 6 7 8 9 Total Points 20 24 35 61 18 20 22 200