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Questions and Answers of
Statistics
1. Two candidates for political office must decide to be for, against, or neutral on a certain referendum. Pollsters have determined that, if candidate R comes out for the referendum, they will gain
1. What do the individual entries of a payoff matrix represent? 2. What is the difference between a pure strategy and a mixed strategy? 3. What is a zero-sum game? 4. Describe the optimal pure
1. When is an entry of a payoff matrix a saddle point? 2. What is a strictly determined game, and what is its value?
What is the expected value of a pair of mixed strategies, and how is it computed?
What is meant by the optimal mixed strategies of R and C, and how are they computed?
State whether or not the games having the given payoff matrices are strictly determined. For those that are, give the optimal pure strategies and the values of the strategies.1.2. 3. 4.
1. Determine the optimal strategy for R for the game with payoff matrix?2. Determine the optimal strategy for C for the game with payoff matrix?
Ruth and Carol play the following game. Each has two cards, a two and a six. Each puts one of her cards on the table. If both put down the same rank, Ruth pays Carol $3. Otherwise, Carol pays Ruth as
An investor is considering purchasing one of three stocks. Stock A is regarded as conservative, stock B as speculative, and stock C as highly risky. If the economic growth during the coming year is
1. Show that, in a two-person game with payoff matrixthe only games that are not strictly determined are those for which either a11 > a12, a11 > a21, a21 or a11 a22, a12 > a22 2. In Check
Assume that the payoff matrix for a two-person game is given byThen assume that R uses a strategy [r 1 - r]. If C chooses column 1, then R can expect a return of a11r + a21 (1 - r). If C chooses
Using the technique of Exercise 17, find a general formula for the best strategy for R in terms of the entries in the payoff matrix.Refer to Exercise 17,Assume that the payoff matrix for a two-person
Determine the expected value of each pair of mixed strategies for the given payoff matrix.1.2. 3. 4.
Determine the optimal strategies for R and for C for the games with the payoff matrices of Exercises 1 and 2?1.2.
1. Explain why 60% of the time, the instructions in Fig. 3 return a 1 and 40% of the time, they return a 2. Type the instructions into your calculator, press the ENTER key 20 times, and count the
Suppose that a game has payoff matrixCalculate the expected values for the following strategies, and determine which of the following four situations is most advantageous to R: (a) R plays [.5
Rework Exercise 9 wit [.3 .7] as Robert's strategy?Refer Exercise 9,Two players, Robert and Carol, play a game with payoff matrix (to Robert)(a) Is the game strictly determined? Why? (b) Suppose
Assume that two players, Renée and Carlos, play a game with the following payoff matrix (to Renée):(a) Is the game strictly determined? Determine the strategy for each player. (b) What
The two players of Exercise 11, Renée and Carlos, play the game again, but this time, the payoff matrix (to Renée) is(a) Is the game strictly determined? Determine the strategy for each
Suppose that, when the offense calls a running play, they gain an average of 1 yard if the defense anticipates a running play, and gain an average of 3 yards if the defense anticipates a passing
Whenever Randy pitches to Chris, he throws either a fastball or a slider. When Randy pitches a fastball, Chris hits the ball with a probability of .25 when he anticipates a fastball, and hits the
Reven and Coddy play a game in which they each simultaneously present a single hand with one or two fingers extended. Reven wins if the total number of fingers extended is even. Otherwise, Coddy
Reven and Coddy play a game in which they each simultaneously present a single hand with one, two, or three fingers extended. Reven wins if the total number of fingers extended is even. Otherwise,
Suppose that a game has payoff matrixCalculate the expected values for the following strategies, and determine which of the following four situations is most advantageous to C: (a) R plays [1 0
1. Refer to Example 3. Suppose that the inspector changes her strategy and visits plant B 80% of the time. How much is Acme's average fine per day? Refer to Example 3, The Acme Chemical Corporation
A small business owner must decide whether to carry flood insurance. She may insure her business for $2 million for $100,000, $1 million for $50,000, or $.5 million for $30,000. Her business is worth
Two players, Robert and Carol, play a game with payoff matrix (to Robert)(a) Is the game strictly determined? Why? (b) Suppose that Robert has strategy [.3 .7]. The opponents agree that the game
Rework Exercise 7 with [.7 .3] as Robert's strategy?Refer to Exercise 7,Two players, Robert and Carol, play a game with payoff matrix (to Robert)(a) Is the game strictly determined? Why? (b)
Two players, Robert and Carol, play a game with payoff matrix (to Robert)(a) Is the game strictly determined? Why? (b) Suppose that Robert has strategy [.7 .3]. The opponents agree that the game
1. Suppose a game has payoff matrixReduce the determination of an optimal strategy for R into a linear programming problem. Just set up the problem, showing the constraints and the objective
Determine the value of the game and the optimal strategy for R.1.2. 3.
Determine the value of the game and the optimal strategy for C.1.2. 3.
A rumrunner attempts to smuggle rum into a country having two ports. Each day, the coast guard is able to patrol only one of the ports. If the rumrunner enters via an unpatrolled port, he will be
Ralph puts a coin in one of his hands, and Carl tries to guess which hand holds the coin. If Carl guesses incorrectly, he must pay Ralph $2. If Carl guesses correctly, then Ralph must pay him $3 if
Suppose that, when the offense calls a running play, they gain an average of 1 yard if the defense anticipates a running play, and gain an average of 3 yards if the defense anticipates a passing
Whenever Randy pitches to Chris, he throws either a fastball or a slider. When Randy pitches a fastball, Chris hits the ball with a probability of .25 when he anticipates a fastball, and hits the
Reven and Coddy play a game in which they each simultaneously present a single hand with one or two fingers extended. Reven wins if the total number of fingers extended is even. Otherwise, Coddy
Reven and Coddy play a game in which they each simultaneously present a single hand with one, two, or three fingers extended. Reven wins if the total number of fingers extended is even. Otherwise,
The Carter Company can choose between two advertising strategies (I and II). Its most important competitor, Rosedale Associates, has a choice of three advertising strategies (a, b, c). The estimated
Use Excel or Wolfram | Alpha to find the expected value and the optimal strategies for the Three-Finger Morra game in Exercise 26. Refer in Exercise 26, Reven and Coddy play a game in which they each
Use Excel or Wolfram | Alpha to find the expected value and the optimal strategies for the Two-Finger Morra game in Exercise 25, Refer to Exercise 25, Reven and Coddy play a game in which they each
1. Rework Exercise 1 for C instead of R.Refer to Exercise 1,Suppose a game has payoff matrixReduce the determination of an optimal strategy for R into a linear programming problem. Just set up the
Determine the value of the game and the optimal strategy for R and for C.1.2. 3.
Give the values of i and n under the given conditions. 1. 3% interest compounded monthly for 2 years 2. 2% interest compounded quarterly for 5 years 3. 2.2% interest compounded semiannually for 20
Solve each problem. 1. Future Value Calculate the future value of $1000 after 2 years if deposited at 2.1% interest compounded monthly. 2. Future Value Calculate the future value of $1000 after 1
1. If you had invested $10,000 on January 1, 2010, at 4% interest compounded quarterly, how much would you have had on January 1, 2016? 2. Ms. Garcia has just invested $100,000 at 2.5% interest
1. Mr. Smith wishes to purchase a $10,000 sailboat upon his retirement in 3 years. He has just won the state lottery and would like to set aside enough cash in a savings account paying 3.4% interest
1. In 1999, the NASDAQ Composite Index grew at a rate of 62.2% compounded weekly. Is this rate better or worse than 85% compounded annually? 2. Would you rather earn 3% interest compounded annually
1. If $2000 is deposited into a savings account earning 1.2% interest compounded quarterly, how much interest is earned during the first quarter year? Second quarter year? Third quarter year? 2. If
1. If $5000 is deposited into a savings account at 1.8% interest compounded monthly, how much interest is earned during the first year? During the second year? 2. If $1000 is deposited for 5 years in
1. How much money must you deposit at 4% interest compounded quarterly in order to earn $406.04 interest in 1 year? 2. Savings Account How much money must you deposit at 2.1% interest compounded
1. Consider the following savings account statement:(a) What interest rate is this bank paying? (b) Give the interest and balance on 4/1/17. (c) Give the interest and balance on 1/1/19. 2. Consider
Concern simple interest. 1. Simple Interest Determine r, n, P, and F for each of the following situations: (a) $500 invested at 1.5% simple interest grows to $503.75 in 6 months. (b) In order to have
1. Calculate the future value after 3 years if $1000 is deposited at 1.2% simple interest. 2. Future Value Calculate the future value after 18 months if $2000 is deposited at 2% simple
1. Determine the (simple) interest rate at which $980 grows to $1000 in 6 months. 2. At what (simple) interest rate will $1000 grow to $1200 in 5 years? 3. How many years are required for $500 to
1. Time Interval Determine the amount of time required for money to double at 2% simple interest. 2. Interest Rate Derive the formula for the (simple) interest rate r at which P dollars grow to F
1. Compute the future value after 1 year for $100 invested at 4% interest compounded quarterly. What simple interest rate will yield the same amount in 1 year? 2. Compute the future value after 1
Calculate the effective rate of interest corresponding to the given nominal rate. 1. 4% interest compounded semiannually 2. 8.45% interest compounded weekly 3. 4.4% interest compounded monthly 4.
1. On January 1, 2014, a deposit was made into a savings account paying interest compounded quarterly. The balance on January 1, 2017 was $10,000, and the balance on April 1, 2017 was $10,100. How
1. If a $1000 investment at compound interest doubles every 6 years, how long will it take the investment to grow to $8000? 2. (True or False) An investment growing at the rate of 12% compounded
1. If the value of an investment grows at the rate of 4% compounded annually for 10 years, then it grows about _______ over the 10-year period. (a) 25% (b) 40% (c) 44% (d) 48% 2. If your stock
1. The same amount of money was invested in each of two different investments on January 1, 2015. Investment A increased by 2.5% in 2015, 3% in 2016, and 8.4% in 2017. Investment B increased by the
Give the values of i, n, P, and F. 1. $500 invested at 2.8% interest compounded annually grows to $558.40 in 4 years. 2. $800 invested on January 1, 2011, at 1.8% interest compounded monthly, grows
1. One thousand dollars is deposited into a savings account at 2.7% interest compounded annually. How many years are required for the balance to reach $1946.53? After how many years will the balance
1. Consider the following two interest options for an investment of $1000: (A) 4% simple interest, (B) 3% interest compounded annually. After how many years will option B outperform option A? 2.
1. What is meant by principal? 2. What is the difference between compound interest and simple interest?
Describe the two types of annuities discussed in this chapter. In each case, identify the present and future values?
1. Give the formula for computing a new balance from a previous balance for each type of annuity? 2. Give a formula relating F and R in an increasing annuity? 3. Give a formula relating P and R in a
1. What are the components of an amortization table of a loan? 2. Give the formula for computing a new balance from a previous balance for a loan?
1. What is a balloon payment? 2. Explain how traditional and Roth IRAs work?
1. How are finance charges on a consumer loan calculated with the add-on method? 2. What are discount points?
1. What is the difference between the effective mortgage rate of a mortgage and the APR? 2. What is an interest-only mortgage?
1. What is an adjustable-rate mortgage? 2. Explain how a sequence of numbers is generated by a difference equation of the form yn = ayn-1 + b, y0 given?
1. What is meant by an initial value for a difference equation? 2. Give the solution of the difference equation yn = ayn-1 + b, y0 given, with a ( 1. With a = 1?
What is meant by the balance in a savings account? Future value?
1. How is the interest rate per period determined from the annual interest rate? 2. Explain how compound interest works?
1. Explain how simple interest works? 2. What is meant by the present value of a sum of money to be received in the future?
Explain the difference between the nominal and effective rates for compound interest?
1. What is an annuity? 2. What is meant by the future value of an annuity? Present value? Rent?
1. If $100 earns 3% interest compounded annually, find the future value after 10 years? 2. Mr. West wishes to purchase a condominium for $240,000 cash upon his retirement 10 years from now. How much
1. A businessman buys a $100,000 piece of manufacturing equipment on the following terms: Interest will be charged at a rate of 4% compounded semiannually, but no payments will be made until 2 years
1. Ms. Jones saved $100 per month for 30 years at 6% interest compounded monthly. How much were her accumulated savings worth? 2. An apartment building is currently generating an income of $2000 per
1. Comparing Investments Investment A generates $1000 at the end of each year for 10 years. Investment B generates $5000 at the end of the fifth year and $5000 at the end of the tenth year. Assume a
1. A person makes an initial deposit of $10,000 into a savings account and then deposits $1000 at the end of each quarter year for 15 years. If the interest rate is 2.2% compounded quarterly, how
1. A savings fund currently contains $300,000. It is decided to pay out this amount with 1.8% interest compounded monthly over a 5-year period. What are the monthly payments? 2. What is the monthly
1. Rework Exercise 26 for a Roth IRA. Refer Exercise 26, Elisa, age 60, is currently in the 30% tax bracket and has $30,000 in a traditional IRA that earns 6% interest compounded annually. She
1. Consider a 15-year mortgage of $90,000 at 6% interest compounded monthly with two discount points and a monthly payment of $759.47. The APR for the mortgage is obtained by solving P = 1 - (1 +
Suppose that a lender gives you a choice between the following two 25-year mortgages of $200,000: Mortgage A: 6.5% interest compounded monthly, two points, monthly payment of $1350.41 Mortgage B: 7%
A 20-year mortgage of $250,000 at 5.75% interest compounded monthly, with two discount points, has a monthly payment of $1755.21. Show that the APR for this mortgage is 6%?
A 30-year mortgage of $100,000 at 5.5% interest compounded monthly, with three discount points, has a monthly payment of $567.79. Assume that the loan is expected to be terminated after 8 years, at
Consider a 25-year mortgage of $380,000 at 6.9% interest compounded monthly, where the loan is interest-only for 10 years. What is the monthly payment during the first 10 years? Last 15 years?
Consider a 25-year $220,000 5/1 ARM with a 2.8% margin and which is based on the CMT index. Suppose the value of the CMT index is 3.5% when the loan is initiated and is 4.55% 5 years later. Assume
Consider the difference equation yn = -3yn-1 + 8, y0 = 1. (a) Generate y1, y2, y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in part (b) to obtain y4?
Consider the difference equation yn = yn-1 - 3/2, y0 = 10. (a) Generate y1, y2, y3 from the difference equation. (b) Solve the difference equation. (c) Use the solution in part (b) to obtain y6?
1. How much money would you have to deposit into a savings account initially at 3.05% interest compounded quarterly in order to have $2474 after 7 years? 2. How much money would you have in the bank
1. How much money must be deposited at the end of each week into an annuity at 2.6% interest compounded weekly in order to have $36,000 after 21 years? 2. Find the monthly payment on a $33,100
1. If you decrease the interest rate for an investment by 10%, will the future value decrease by 10%? 2. If interest is compounded semiannually, will the effective rate be higher or lower than the
Consider a decreasing annuity. If the amount withdrawn each month increases by 5%, will the duration decrease by 5%? Give an example to justify your answer.
Give an intuitive explanation for why successive payments for a mortgage contribute steadily more toward repayment of the principal?
1. Which is a better investment: 3% compounded annually or 2.92% compounded daily? 2. Ms. Smith deposits $200 at the end of each month into a bond fund yielding 3% interest compounded monthly. How
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