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Statistics Probability Inference And Decision 1st Edition Robert L. Winkler, William L. Hays - Solutions

11. Suppose that you are offered a choice between Bets A and B:Bet A: You win $1,000,000 with certainty (that is, with probability 1).Bet B: You win $5,000,000 with probability .1 You win $1,000,000 with probability .89 You win $0 with probability .01.
10. You must choose between two acts, where the payoff matrix is as follows(in terms of dollars):
9. Why is the maximization of expected monetary value (that is, EP with the payoffs expressed in terms of money) not always a reasonable criterion for decision-making? What problems does this create for the decision maker?
8. In Exercise 3, if P (I) = .25, P (B) = .45, and P (C) = .30, find the expected payoff and expected loss of each of the three actions. Using EP (or EL) as a decision-making criterion, which action is optimal?
7. In Exercise 2, find the expected payoff and the expected loss of each action, and find the action which has the largest expected payoff and smallest ex¬pected loss, given that the probabilities of the various states of the world are P(A) = .10, P(B) = .20, P(C) = .25, P{D) = .10, and P(E) = .35.
6. What is the primary disadvantage of decision-making rules such as maximin, maximax, and minimax loss?
5. For the payoff and loss tables in Exercise 3, find the action which is optimal under the maximin, maximax, and minimax loss rules.
4. For the payoff and loss tables in Exercise 2, find the action which is optimal under each of the following rules:(a) maximin(b) maximax(c) minimax loss.
3. Consider the following loss table:
2. You are given the following payoff table:
1. Explain the difference between decision-making under certainty and decision¬making under uncertainty.
47. Consider a situation in which you and a friend both must choose a number from the two numbers 1 and 2. The relevant payoffs are as follows:If you choose 1 and he chooses 1, If you choose 1 and he chooses 2, If you choose 2 and he chooses 1, If you choose 2 and he chooses 2, you both win $10 he
46.Explain how game theory problems (that is, decision-making problems in which there is some “opponent”) can be analyzed using the same techniques which are used in decision-making under uncertainty. Instead of “states of the world,” we have “actions of the opponent.” Might this make
45. Suppose that an investor has a choice of four investments (assume that he must choose one, and only one, of the four—he cannot divide his money among them):A: a savings account in a bank.B: a stock which moves counter to the general stock market.C: a growth stock wThich moves at a faster pace
44. In Exercise 42, find the EVSI and ENGS for samples of size 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, assuming that the cost of sampling is 2. On a graph, draw curves representing EVSI, ENGS, and CS.
43. The payoff in a certain decision-making problem depends on p, the parameter of a Bernoulli process. The payoff functions are State of the World 70R-30B 70B-30R 6 -2-6 10 and P(ai, p) = —10 + lOOp P(a2, p) = 50p.
42. Suppose that the payoff functions of two actions are P (oi, y) = 50 + .5y and P(a2, y) = 70— .5/x, where y is the mean of a normal process with variance 1200.(a) Find the breakeven value, /i&.(b) If the prior distribution is a normal distribution with mean M' — 25 and variance
41. In Exercise 36, show that the EVPI is equal to the expected loss of the optimal action under the prior distribution.
40. Do Exercise 39 with the following payoff table (in dollars):70R-30B Your Guess 70R-30B
39. Consider a bookbag filled with 100 poker chips. You know that either 70 of the chips are red and the remainder blue, or that 70 are blue and the remainder red. You must guess whether the bookbag has 70R-30B or 70B-30R. If you guess correctly, you win $5. If you guess incorrectly, you lose $3.
38. In Exercise 37, suppose that you also wanted to consider samples of other sizes.(a) Find EVSI for a sample of size 2.(b) Find EVSI for a sample of size 5.(c) Find EVSI for a sample of size 10.(d) If the cost of sampling is .50 per unit sampled, find the expected net gain of sampling (ENGS) for
37. In Exercise 36, suppose that sample information is available in the form of a random sample of consumers. For a sample of size one,(a) find the posterior distribution if the one person sampled will purchase the item, and find the value of this sample information(b) find the posterior
36.Astore must decide whether or not to stock a new item. The decision depends on the reaction of consumers to the item, and the payoff table is as follows:
35. In Exercise 8, find the expected value of perfect information. If you could purchase perfect information for $5 (assuming the values in the table are given in terms of dollars), would you do so?
34. In Exercise 12, what is the value of perfect information that there is oil?What is the value of perfect information that there is no oil? From these results, calculate the expected value of perfect information in this example.
33. What is the difference between a terminal decision and a preposterior decision?Are they at all related?
32. Carefully distinguish between “classical” inferential statistics, Bayesian in¬ferential statistics, and decision theory.
31. Compare and contrast the determination of a and (3 in hypothesis testing prob¬lems according to (a) the classical “convention” presented in Chapter 7 and (b) the decision-theoretic approach, as illustrated in Exercise 30.
30. Consider the hypotheses Hq:h = 10 and Hi’.y. = 12, where r is the mean of a normally distributed population with variance 1. The prior distribution is diffuse (take a normal distribution with N' = 0), and a sample of size 10 is to be taken. If L\ = 100 and Ln = 50,(a) find the region of
29. For a sample of size one from a normal population with known variance 25, show that the classical test of Ho-n = 50 versus Hi’./jl — 60 with a — .05 is a likelihood ratio test. That is, show that the rejection region can be expressed in the form LR k, where k is some constant, since you
28.Astatistician is interested in testing the hypotheses and H0:p> 120 H^.ft < 120
27. In Exercise 24, Chapter 8, the marketing manager decides that the loss which will be suffered if the company markets the product and p is in fact only .10 is three times as great as the loss which will be suffered if the company fails to market the product and p is actually .20. Should the
26. In Exercise 11, Chapter 8, the production manager must make an estimate of the mean weight of items turned out by the process in question. His loss function is linear with k0 = ku. What should his estimate be?
25. In Exercise 7, Chapter 8, determine an estimate of p from the posterior dis¬tribution if the loss function is L(a, p) = k(a — p)2.How does this differ from the estimate obtained from the prior distribution using the same loss function? From the estimate obtained from the sample alone?
24. If the loss function in a point estimation problem is of the form L (a, 6)0 1if | a — 9 | < k, otherwise, where k is some very small positive number, what would the optimal estimate be? Can you think of any realistic situations in which the loss function might be of this form?
23. If a statistician wishes to estimate a parameter d subject to a loss function which is linear with k0 = 2kU) and his distribution of 6 is an exponential distribution with X = 4, what is the optimal estimate of 0?
22. If a person faces a point estimation problem with a linear loss function with ku — 4 and k0 = 3, does he need to assess an entire probability distribution or can he determine a certainty equivalent? Explain.
21. Comment on the following statement: “In taking the sample mean as an estimator of the population mean, the statistician is acting essentially as though he had a quadratic loss function.” Explain the difference between the approach to point estimation taken in Chapter 6 and the approach
20. In Exercises 15, 16, 17, and 19, what would the optimal estimates be if the loss function were quadratic instead of linear [that is, if the loss function were of the form (9.13.1)]?
19. An economist is asked to predict a future value of a particular economic indicator, and he thinks that his loss function is linear with ku = 4k0. If his distribution is a uniform distribution on the interval from 650 to 680, what should his estimate be?
18. Prove that if a decision maker’s loss function in a point estimation problem is given by Equation (9.12.1), then the kj (ku + k0) fractile of the decision maker’s distribution of 6 is the optimal point estimate.
17. A sales manager is asked to forecast the total sales of his division for a forth¬coming period of time. He feels that his loss function is linear as a function of the difference between his estimate and the true value, but he also feels that an error of overestimation is three times as serious
16. A hot dog vendor at a football game must decide in advance how many hot dogs to order. He makes a profit of $.10 on each hot dog which is sold, and he suffers a $.20 loss on hot dogs which are unsold. If his distribution of the number of hot dogs that will be demanded at the football game is a
15. Suppose that a contractor must decide whether or not to build any speculative houses (that is, houses that he would have to find a buyer for), and if so,
14. Attempt to determine your own utility function for money in the range from —$500 to +$500. If you were given actual decision-making situations, would you act in accordance with this utility function?
13. Comment on the following statement from the text: “Some individuals appear to be risk-takers for some decisions (such as gambling) and risk-avoiders for other decisions (such as purchasing insurance).” Can you explain why this phenomenon occurs? Can you draw a utility function which would
12. Suppose that you are contemplating drilling an oil well, with the following payoff table (in terms of thousands of dollars):
11. Suppose that you are offered a choice between Bets A and B:Bet A: You win $1,000,000 with certainty (that is, with probability 1).Bet B: You win $5,000,000 with probability .1 You win $1,000,000 with probability .89 You win $0 with probability .01.
10. You must choose between two acts, where the payoff matrix is as follows(in terms of dollars):
9. Why is the maximization of expected monetary value (that is, EP with the payoffs expressed in terms of money) not always a reasonable criterion for decision-making? What problems does this create for the decision maker?
8. In Exercise 3, if P (I) = .25, P (B) = .45, and P (C) = .30, find the expected payoff and expected loss of each of the three actions. Using EP (or EL) as a decision-making criterion, which action is optimal?
7. In Exercise 2, find the expected payoff and the expected loss of each action, and find the action which has the largest expected payoff and smallest ex¬pected loss, given that the probabilities of the various states of the world are P(A) = .10, P(B) = .20, P(C) = .25, P{D) = .10, and P(E) = .35.
6. What is the primary disadvantage of decision-making rules such as maximin, maximax, and minimax loss?
5. For the payoff and loss tables in Exercise 3, find the action which is optimal under the maximin, maximax, and minimax loss rules.
4. For the payoff and loss tables in Exercise 2, find the action which is optimal under each of the following rules:(a) maximin(b) maximax(c) minimax loss.
3. Consider the following loss table:
2. You are given the following payoff table:
1. Explain the difference between decision-making under certainty and decision¬making under uncertainty.
24. Suppose that a marketing manager is interested in p, the proportion of con¬sumers that will buy a particular new product. He considers the following two hypotheses:H0:p = .10 Hi'.p = .20.His prior probabilities are P (p = .10) = .85 and P (p = .20) = .15, and a random sample of 8 consumers
Follow the procedure in Exercise 39 for the variable D, the daily change in the price of one share of IBM common stock?
Try to assess a probability distribution for T, the maximum temperature tomorrow in the city where you live. In assessing the distribution, use two or three of the methods proposed in the text and compare the results. Then, save the distribution and look at it again after you find out the true
Exercise 37 suggests a Bayesian approach to inferences regarding the differ¬ence between two means, under the conditions that the two populations of interest are normally distributed and the variances are known. Using the notation of Exercise 37, determine a general formula for finding the
Suppose that a statistician is interested in the difference in the means of two normal populations, pi and p2. Let p = pi — p2. Each population has vari¬ance 100, and the two populations are independent. A sample of size 25 is taken from the first population, with sample mean Mi = 80, and a
Suppose that X is normally distributed with mean /x and variance a2 = 400.The prior distribution of n is normally distributed with mean M' — — 60 and variance e/2 = 40. Furthermore, a sample of size 20 is taken, with sample mean M = —69.(a) Find the posterior distribution of /x.(b) Find a 68
Suppose that a statistician is interested in H0:/x = 50 and Hi'.fx 9^ 50. His prior distribution consists of a mass of probability of .25 at /x = 50, with the remaining .75 of probability distributed uniformly over the interval from H = 40 to /x — 60. Find the prior probability that:(a) 45 < ix <
In Exercise 11, suppose that the manager is interested in the hypotheses Ho'-ix = 110 and Hi :/x ^ 110. From the posterior distribution, what is the pos¬terior odds ratio of Ho to Hj? What is the posterior odds ratio if the hypo¬theses are #0:109 < fx < 111 and Hi'.fx < 109 or /u > 111?
An automobile manufacturer claims that the average mileage per gallon of gas for a particular model is normally distributed with M' = 20 and a' = 4, provided that the car is driven on a level road at a constant speed of 30 miles per hour. A rival manufacturer decides to use this as a prior
Comment on the following statement: “A hypothesis such as /x = 100 is not realistic and should be modified somewhat; otherwise, the Bayesian approach to hypothesis testing may not be applicable.”
In testing a hypothesis concerning the mean of a normal process with known variance against a one-tailed alternative, discuss the relationship between the classical p-value and the posterior probability P" (Ho), (a) if the prior distribution is diffuse, (b) if the prior distribution is not diffuse.
Prove that the posterior odds ratio of one hypothesis to another is equal to the product of the prior odds ratio and the likelihood ratio.
A statistician is interested in the mean p of a normal population, and his prior distribution is normal with mean 800 and variance 12. The variance of the population is known to be 72. How large a sample would be needed to guar¬antee that the variance of the posterior distribution is no larger
In Exercise 11, the manager is interested in the hypotheses Ho'-p. < 110 and Hi'.p > 110. From the prior distribution, find the prior odds ratio; also, from the posterior distribution, find the posterior odds ratio. If the losses involved in a decision-making problem involving the production
In Exercise 24, it would be more realistic to consider hypotheses such as the following:Ho'-p < .15 H1:p> .15.If the prior distribution is a beta distribution with r' = 1 and N' = 9, find the posterior distribution. How could you use the posterior distribution to find the posterior odds ratio of Ho
Discuss: “The Bayesian approach to statistics can be thought of as an exten¬sion of the classical approach.”
Suppose that a marketing manager is interested in p, the proportion of con¬sumers that will buy a particular new product. He considers the following two hypotheses:H0:p = .10 Hi'.p = .20.His prior probabilities are P (p = .10) = .85 and P (p = .20) = .15, and a random sample of 8 consumers results
The random variable X is normally distributed with unknown mean ji and known variance
In Exercise 11, find(a) a point estimate for /x based on the prior distribution alone(b) a point estimate for n based on the sample information alone(c) a point estimate for fx based on the posterior distribution(d) a 90 percent credible interval for /x based on the prior distribution alone(e) a 90
Carefully distinguish between a classical confidence interval and a Bayesian credible interval, and also between the classical and Bayesian approaches to point estimation.
The beta distribution with r' = N' = 0 and the beta distribution with r' = 1 and N' — 2 have both been used by statisticians as “diffuse” beta distribu¬tions. Discuss the advantages and disadvantages of each.
Give a few examples of situations in which your prior distribution would effectively be diffuse, and a few examples in which it would definitely not be diffuse relative to a given sample.
Comment on the following statement: “An informationless, or diffuse, prior state of information is not informationless in the usual meaning of the word, but rather in a relative sense.”
Comment on the following statement: “We cannot speak of sensitivity except in connection with a particular decision-making situation.” Can you think of an example in which the decision-making procedure would be quite in¬sensitive to changes in the prior distribution? An example in which it
In assessing a distribution for the mean height of a certain population of college students, a person decides that his distribution is normal, the median is 70 inches, the .20 fractile is 67 inches, and the .80 fractile is 72 inches.Can you find a normal distribution with these fractiles? Comment
Suppose that a statistician decides that his prior distribution for a certain parameter 6 is an exponential distribution and that the .67 fractile of this distribution is 2. Find the exact form of the distribution.
Explain the parametrization of a normal prior distribution in terms of N'and M', as given in Section 8.12. How does this parametrization help us to see the relative weights of the prior and the sample information in computing the mean of the posterior distribution? For the prior and posterior
In reporting the results of a statistical investigation, a statistician reports that his posterior distribution for p is a normal distribution with mean 52 and variance 10, and that his sample of size 4 with sample mean 55 was taken from a normal population with variance 100. On the basis of this
You are attempting to assess a prior distribution for the mean of a process, and you decide that the .25 fractile if your distribution is 160 and the .60 fractile is 180. If your prior distribution is normal, determine the mean and variance.
A production manager is interested in the mean weight of items turned out by a particular process. He feels that the weight is normally distributed with standard deviation 2, and his prior distribution for p is a normal distribution with mean 110 and variance .4. He randomly selects five items from
Try to assesses a subjective distribution of p, the probability of rain tomorrow.Can you find a beta distribution which expresses your judgments reasonably well?
In sampling from a Bernoulli process, the posterior distribution is the same whether we sample with N fixed (binomial sampling) or with r fixed (Pascal sampling). Explain why this is true. Suppose the statistician merely sampled until he was tired and decided to go home—would the posterior
In Exercise 2, suppose that you feel that in comparing p = .6 and p = .4, the odds are 3-to-2 in favor of p — .6. Furthermore, in comparing p = A and p = .5, the odds are 5-to-l in favor of p = .5. On the basis of these odds, find your prior distribution for p. Does this suggest one possible
In Exercise 2, suppose that you feel that the mean of your prior distribution is 1/2 and the variance of the distribution is 1/20. If your prior distribution is a member of the beta family, find r' and N' and determine the posterior distribution following the sample of size 6.
In Exercise 1, suppose that the prior distribution could have been represented by a beta distribution with r' — 4 and N' = 10. Find the posterior distri¬bution. Also, find the posterior distribution corresponding to the following beta prior distributions:(a) r' = 2 N' = 5(b) / = 8 N' = 20(c) r'
Discuss the importance of conjugate families of distributions in Bayesian statistics.
In Bayes’ theorem, why is it necessary to divide by 2 (prior probability) (like¬lihood) in the discrete case and by / (prior density) (likelihood) in the con¬tinuous case?
Explain the statement, “the terms prior and posterior, when applied to prob¬abilities, are relative terms, relative to a given sample.” Is it possible for a set of probabilities to be both prior and posterior probabilities? Explain.
You feel that the probability of a head on a toss of a particular coin is either.4, .5, or .6. Your prior probabilities are P (A) = .1, P(.5) = .7, P(.6) = .2.You toss the coin three times and obtain one head and two tails. What are the posterior probabilities? If you then toss the coin three more
Suppose that it is believed that the proportion of consumers in a large popu¬lation that will purchase a certain product is either .2, .3, .4, or .5. Further¬more, the prior probabilities for these four values are P(.2) = .2, P(.3) = .3, P(A) = .3, and P (.5) = .2. A sample of size 10 is taken
Discuss the importance of the normality assumption in each of the tests discussed in this chapter which involve such an assumption. Is the assumption more crucial for some tests than for others? What other assumptions are important?
Suppose that two independent random samples were to be compared in terms of their variability. If the values in one of these samples were multiplied by the constant k, what would be the effect on the F-ratio based on the variances of the two samples? State whether this suggests a way to test a

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