Chapter 5 Solutions Manual sta1610 2021 PDF

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For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, StatisticsChapter 5: Discrete Probability SolutionsTextbook Exercises: 1 Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of heads...

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Chapter 5: Discrete Probability Solutions

Textbook Exercises: 1 Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of heads occurring on the two tosses. c. Show what value the random variable would assume for each of the experimental outcomes. d. Is this random variable discrete or continuous? 2 Consider the experiment of a worker assembling a product. a. Define a random variable that represents the time in minutes required to assemble the product. b. What values may the random variable assume? c. Is the random variable discrete or continuous?

3 Three students have interviews scheduled for summer employment. In each case the interview results in either an offer for a position or no offer. Experimental outcomes are defined in terms of the results of the three interviews. a. List the experimental outcomes. b. Define a random variable that represents the number of offers made. Is the random variable continuous? c. Show the value of the random variable for each of the experimental outcomes.

4 Suppose we know home mortgage rates for 12 Danish lending institutions. Assume that the random variable of interest is the number of lending institutions in this group that offers a 30-year fixed rate of 1.5 per cent or less. What values may this random variable assume?

5 To perform a certain type of blood analysis, lab technicians must perform two procedures. The first procedure requires either 1 or 2 separate steps, and the second procedure requires either 1, 2 or 3 steps. a. List the experimental outcomes associated with performing the blood analysis. b. If the random variable of interest is the total number of steps required to do the complete analysis (both procedures), show what value the random variable will assume For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

for each of the experimental outcomes. 6 Listed is a series of experiments and associated random variables. In each case, identify the values that the random variable can assume and state whether the random variable is discrete or continuous.

7 The probability distribution for the random variable X follows.

a. Is this probability distribution valid? Explain. b. What is the probability that X = 30? c. What is the probability that X is less than or equal to 25? d. What is the probability that X is greater than 30?

8 The following data were collected by counting the number of operating rooms in use at a general hospital over a 20-day period. On three of the days only one operating room was used, on five of the days two were used, on eight of the days three were used, and on four days all four of the hospital’s operating rooms were used. a. Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day. b. Draw a graph of the probability distribution. c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution. For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

9 A technician services mailing machines at companies in the Berne area. Depending on the type of malfunction, the service call can take 1, 2, 3 or 4 hours. The different types of malfunctions occur at about the same frequency. a. Develop a probability distribution for the duration of a service call. b. Draw a graph of the probability distribution. c. Show that your probability distribution satisfies the conditions required for a discrete probability function. d. What is the probability a service call will take three hours? e. A service call has just come in, but the type of malfunction is unknown. It is 3:00 p.m. and service technicians usually finish work at 5:00 p.m. What is the probability the service technician will have to work overtime to fix the machine today?

10 In a monthly charity competition, CAD$10,000 is awarded for every 2,666,667 bonds (each costing CAD$1) in the competition. The prize money is divided into 295 prizes with the following values:

Number CAD$

of prizes

1,000

1

500

1

250

2

100

3

50

20

25

268

a. What is the probability of winning a prize with one bond in each draw? b. If a prize is won, what is the probability that it is one for CAD$100 or more? c. What is the expected gain each month for each bond? d. What annual rate of interest does this correspond to?

11 A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2 or 3. Let X be a random variable indicating the number of sessions required to gain the patient’s trust. The following probability function has been proposed. p(x) = x For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

6 for x = 1, 2, or 3 a. Is this probability function valid? Explain. b. What is the probability that it takes exactly two sessions to gain the patient’s trust? c. What is the probability that it takes at least two sessions to gain the patient’s trust?

12 The following table is a partial probability distribution for the MRA Company’s projected profits (X = profit in €’000s) for the first year of operation (the negative value denotes a loss).

a. What is the proper value for p(200)? What is your interpretation of this value? b. What is the probability that MRA will be profit table? c. What is the probability that MRA will make at least €100,000?

13 The following table provides a probability distribution for the random variable X.

a. Compute E(X), the expected value of X. b. Compute σ 2, the variance of X. c. Compute σ , the standard deviation of X.

14 The following table provides a probability distribution for the random variable Y.

For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

a. Compute E(Y). b. Compute Var(Y) and σ .

15. The table below summarizes the joint probability distribution for the percentage monthly return for two ordinary shares 1 and 2. In the case of share 1, the % return X has historically been -1, 0 or 1. Correspondingly, for share 2, the % return Y has been -2, 0 or 2.

Table 5.6 Percent monthly return probabilities for shares 1 and 2 %

share 2

Monthly

Y

return

-2

0

2

-1

0.1

0.1

0.0

0

0.1

0.2

0.0

1

0.0

0.1

0.4

share 1 X

a. Determine E(Y), E(X) Var(X) and Var(Y) 16. Odds in horse race betting are defined as follows: 3/1 (three to one against) means a horse is expected to win once for every three times it loses; 3/2 means two wins out of five races; 4/5 (five to four on) means five wins for every four defeats, etc.

a. Translate the above odds into ‘probabilities’ of victory. b. In the 2.45 race at L’Arc de Triomphe the odds for the five runners were:

Phillipe Bois 1/1 Gallante Effort 5/2 Satin Noir 11/2 Victoire Antheme 9/1 Comme Rambleur 16/1 For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

Calculate the ‘probabilities’ and their sum. c. How much would a bookmaker expect to profit in the long run at such odds if it is assumed each horse is backed equally? (Hint: Assume the true probabilities are proportional to the ‘probabilities’ just calculated and consider the payouts corresponding to a notional :1 wager being placed on each horse.) d. What would the bookmaker’s expected profit have been if horses had been backed in line with the true probabilities?

17. A certain machinist works an eight-hour shift. An efficiency expert wants to assess the value of this machinist where value is defined as value added minus the machinist’s labour cost. The value added for the work the machinist does is €30 per item and the machinist earns €16 per hour. From past records, the machinist’s output per shift is known to have the following probability distribution:

a. What is the expected monetary value of the machinist to the company per shift? b. What is the corresponding variance value?

18. A company is contracted to finish a €100,000 project by 31 December. If it does not complete on time a penalty of €8,000 per month (or part of a month) is incurred. The company estimates that if it continues alone there will be a 40 per cent chance of completing on time and that the project may be one, two, three or four months late with equal probability.

Subcontractors can be hired by the firm at a cost of €18,000. If the subcontractors are hired then the probability that the company completes on time is doubled. If the project is still late it will now be only one or two months late with equal probability. a. Determine the expected profit when (i) subcontractors are not used (ii) subcontractors are used b. Which is the better option for the company?

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19. A typical slot machine has 3 dials, each with 20 symbols (cherries, plums, lemons, oranges, bells and bars). A set of dials is configured as follows:

Dial 1

2

3

Cherries

7

7

0

Oranges

3

7

6

Lemons

3

0

4

Plums

4

1

6

Bells

2

2

3

Bars

1

3

1

20

20

20

According to this table, of the 20 slots on dial 1, 7 are cherries, 3 oranges, etc. Payoffs (€) for a €1 bet are given below:

Dial 1

2

3

Payoff (€)

Bar

Bar

Bar

60

Bell

Bell

Bell

20

Bell

Bell

Bar

18

Plum

Plum

Plum

14

Orange

Orange

Orange

10

Orange

Orange

Bar

8

Cherry

Cherry

Anything

4

Cherry

No cherry

Anything

2

Anything

-1

else

a. Compute the player’s expected winnings on a single play of the slot machine. (Assume that each dial acts independently.)

20. The demand for a product of Cobh Industries varies greatly from month to month. The probability distribution in the following table, based on the past two years of data, shows the company’s monthly demand.

For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

a. If the company bases monthly orders on the expected value of the monthly demand, what should Cobh’s monthly order quantity be for this product? b. Assume that each unit demanded generates €70 in revenue and that each unit ordered costs €50. How much will the company gain or lose in a month if it places an order based on your answer to part (a) and the actual demand for the item is 300 units?

21. Given below is a bivariate distribution for the random variables X and Y taking the values x and y respectively.

p(x,

x

y

0.2

50

80

0.5

30

50

0.3

40

60

y)

a. Compute the expected value and the variance for X and Y. b. Develop a probability distribution for X + Y. c. Using the result of part (b), compute E(X + Y) and Var (X + Y). d. Compute the covariance and correlation for X and Y . Are X and Y

positively correlated, negatively correlated, or uncorrelated? e. Is the variance of the sum of x and y bigger, smaller, or the same

as the sum of the individual variances? Why?

22. A person is interested in constructing a portfolio. Two stocks are being considered. Let X = per cent return for an investment in stock 1, and Y = per cent return for an investment in stock 2. The expected return and variance for stock 1 are E(X) = 8.45% and Var( X) = 25. The expected return and variance for stock 2 are E(Y) = 3.20% and Var(Y) = 1. The covariance between the returns is sXY = −3. a. What is the standard deviation for an investment in stock 1 and for

an investment in stock 2? Using the standard deviation as a measure of risk, which of these stocks is the riskier investment? For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

b. What is the expected return and standard deviation, in dollars, for a person

who invests $500 in stock 1? c. What is the expected per cent return and standard deviation for a

person who constructs a portfolio by investing 50% in each stock? d. What is the expected per cent return and standard deviation for a

person who constructs a portfolio by investing 70% in stock 1 and 30% in stock 2? e. Compute the correlation coefficient for x and y and comment on

the relationship returns for the two stocks.

23. The Chamber of Commerce in a Canadian city has conducted an evaluation of 300 restaurants in its metropolitan area. Each restaurant received a rating on a 3-point scale on typical meal price (1 least expensive to 3 most expensive) and quality (1 lowest quality to 3 greatest quality). A cross-tabulation of the rating data is shown below. Forty-two of the restaurants received a rating of 1 on quality and 1 on meal price, 39 of the restaurants received a rating of 1 on quality and 2 on meal price, and so on. Forty-eight of the restaurants received the highest rating of 3 on both quality and meal price. Meal price (Y) Quality

1

2

3

Total

1

42

39

3

84

2

33

63

54

150

3

3

15

48

66

78

117

105

300

(X)

Total

a. Develop a bivariate probability distribution for quality and meal price of a randomly selected restaurant in this Canadian city. Let X = quality rating and Y = meal price. b. Compute the expected value and variance for quality rating, X. c. Compute the expected value and variance for meal price, Y. d. The Var(X + Y) = 1.6691. Compute the covariance of X and Y. What can you say about the relationship between quality and meal price? Is this what you would expect? e. Compute the correlation coefficient between quality and meal price. What is the strength of the relationship? Do you suppose it is likely that a lowFor use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

cost restaurant in this city that is also high quality can be found? Why or why not?

24. J.P. Morgan Asset Management publishes information about financial investments. Over the past 10 years, the expected return for the S&P 500 was 5.04 per cent with a standard deviation of 19.45 per cent and the expected return over that same period for a core bonds fund was 5.78 per cent with a standard deviation of 2.13 per cent (J.P. Morgan Asset Management, Guide to the Markets, 1st Quarter, 2012). The publication also reported that the correlation between the S&P 500 and core bonds is −.32. You are considering portfolio investments that are composed of an S&P 500 index fund and a core bonds fund. a. Using the information provided, determine the covariance between the S&P 500 and core bonds. b. Construct a portfolio that is 50 per cent invested in an S&P 500 index fund and 50 per cent in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? c. Construct a portfolio that is 20 per cent invested in an S&P 500 index fund and 80 per cent invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? d. Construct a portfolio that is 80 per cent invested in an S&P 500 index fund and 20 per cent invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio? e. Which of the portfolios in parts (b), (c) and (d) has the largest expected return? Which has the smallest standard deviation? Which of these portfolios is the best investment alternative? f. Discuss the advantages and disadvantages of investing in the three portfolios in parts (b), (c) and (d). Would you prefer investing all your money in the S&P 500 index, the core bonds fund, or one of the three portfolios? Why?

25 Consider a binomial experiment with two trials and π = 0.4. a. Draw a tree diagram for this experiment (see Figure 5.3). b. Compute the probability of one success, p(1). c. Compute p(0). d. Compute p(2). e. Compute the probability of at least one success. f. Compute the expected value, variance, and standard deviation.

26 Consider a binomial experiment with n = 10 and π = 0.10. For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020

a. Compute p(0). b. Compute p(2). c. Compute P(X ≤ 2). d. Compute P(X ≥ 1). e. Compute E(X). f. Compute Var(X) and σ .

27 Consider a binomial experiment with n = 20 and π = 0.70. a. Compute p(12). b. Compute p(16). c. Compute P(X ≥ 16). d. Compute P(X ≤ 15). e. Compute E(X). f. Compute Var(X) and σ .

28 Refer to Exercise 10. An investor wishes to buy enough bonds to ensure there is a 1 in 10 chance of winning at least one prize in each monthly draw. How many bonds should be bought?

29 It takes at least 9 votes from a 12 member jury to convict a defendant. Suppose that the probability that a juror votes a guilty person innocent is 0.2 whereas the probability that the juror votes an innocent person guilty is 0.1. a. If each juror acts independently and 65% of defendants are guilty, what is the probability that the jury renders a correct decision. b. What percentage of defendants is convicted? 30 A firm bills its accounts at a 1 per cent discount for payment within ten days and the full amount is due after ten days. In the past 30 per cent of all invoices have been paid within ten days. If the firm sends out eight invoices during the first week of January, what is the probability that: a. No one receives the discount? b. Everyone receives the discount? c. No more than three receive the discount? d. At least two receive the discount?

31 In a game of 'Chuck a luck' a player bets on one of the numbers 1 to 6. Three dice are then rolled and if the number bet by the player appears i times (i=1,2,3) the player then wins For use with Anderson, Sweeney, Williams...