1. Baseball America has noticed the number of homeruns has been increasing in recent years in the MLB. They want to develop a 95% confidence interval that captures the true home run percentage. Home run percentage is defined as the number of home runs per 100 at bats. To do so, they randomly selected 64 current MLB players and calculated their homeruns per at bat for the previous year, and obtained a sample mean and sample standard deviation of 2.2 and 1.7, respectively.

a. Compute a 95% confidence interval for the population mean ? of the home run rate for all MLB players. Interpret with context to the problem.

b. The home run percentages for three MLB players are:

• Player 1: Primetime Peanuts: 2.1

• Player 2: Spleens “No Pop” McGillicuddy: 4

• Player 3: Big Dog Lebowski: 1.5

Assess the confidence interval you calculated and describe how the home run rate for these three players compare to the interval calculated for the population mean.


c. If the confidence level was increased to 99%, would the interval be wider or narrower? Why?

d. Before collecting any data, Baseball America wants to achieve a maximum bound on error of 0.3. They suspect the range of home run rates to be 1.5 to 8. How large a sample should be used to be 95% confident of achieving this level of accuracy?

PLEASE SHOW ALL FORMULA AND WORK.

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A global equity manager is assigned to select stocks from a universe of large stocks throughout the world. The manager will be evaluated by comparing her returns to the return on the MSCI World Market Portfolio, but she is free to hold stocks from various countries in whatever proportions she finds desirable. Results for a given month are contained in the following table:

a. Calculate the total value added of all the manager’s decisions this period. (Do not round intermediate calculations. Round your answer to 2 decimal places. Negative amount should be indicated by a minus sign.)

Added Value=

b. Calculate the value added (or subtracted) by her country allocation decisions. (Do not round intermediate calculations. Round your answer to 2 decimal places. Negative amount should be indicated by a minus sign.)

Contribution of country allocation=

c. Calculate the value added from her stock selection ability within countries. (Do not round intermediate calculations. Round your answer to 2 decimal places. Negative amount should be indicated by a minus sign.)

Contribution of stock selection=

PLEASE READ: This is one question with 3 parts to it, please answer the full question.

Mark M. Upp has just been fired as the university bookstore manager for setting prices too low (only 20 percent above suggest retail). He is considering opening a competing bookstore near the campus, and he has begun an analysis of the situation. There are two possible sites under consideration. One is relatively small, while the other is large. If he opens at Site 1 and demand is good, he will generate a profit of $ 200,000. If demand is low, he will lose $180,000. If he opens at Site 2 and demand is high, he will generate a profit of $100,000, but he will lose $20,000 if demand is low. He also has the option of not opening either. He believes that there is a 50 percent chance that demand will be high. Mark can purchase a market research study from Brooklyn College. The survey costs $10,000. The probability of a good demand given a favorable study is 0.8. The probability of a good demand given an unfavorable study is 0.3. There is a 45 percent chance that the study will be favorable. Draw a decision tree to determine the following:

a)What should Mark’s decisions be?

b)What is the maximum amount Mark should be willing to pay for this study?

c)What is the efficiency of the study?

Hint: The revised probabilities have already been calculated for you.

Lola must decide on a price for her homemade aromatherapy candles. The number of candles she expects to sell depends on the price that is set by her competitor, Sunny’s Scents of Serenity. Lola must set her price before she knows what Sunny will do. Lola believes that Sunny’s price is a random variable C having the following probability mass function. P[C =$8]=0.4,P[C=$10]=0.3,P[C=$12]=0.2,P[C=$15]=0.1. IfLolachargesaprice p1 and Sunny charges a price p2, Lola sells 20 + 5(p2 – p1) candles. Lola is considering charging $6, $10, or $12 for her candles. It costs her $1 in time and materials to make each candle.

a. Under the Expected Monetary Value criterion, which price should Lola charge?

b. Lola can bribe Sunny’s boyfriend to tell her what price she (Sunny) plans to charge. At most how much should Lola be willing to pay for this information?

c. Lola is not comfortable with the payout she gets ($405) when she sets her price to $10 and Sunny sets hers to $15. She think it should be higher. How large must it become before the option of setting her price to $10 become optimal on expected monetary value (EMV) grounds?

1. (15 pts) The following data show the percentage change in population for the 50 states and the District of Columbia from 2000 to 2009.

a. Construct a relative frequency distribution (5 pts) and draw a histogram of the data. (4 pts)

b.  Round the data to integers in percentage and then create a stem-and-leaf display of these data. (4 pts)

c.  Describe the shape of the distribution. (2 pts)

You are a banker and are confronted with a pool of loan applicants, each of whom can be either low risk or high risk. There are 600 low-risk applicants and 400 highrisk applicants and each applicant is applying for a $100 loan. A low-risk borrower will invest the $100 loan in a project that will yield $150 with probability 0.8 and nothing with probability 0.2 one period hence. A high-risk borrower will invest the $100 loan in a project that will yield $155 with probability 0.7 and nothing with probability 0.3 one period hence. You know that 60% of the applicant pool is low risk and 40% is high risk, but you cannot tell whether a specific borrower is low risk or high risk. You are a monopolist banker and have $50,000 available to lend. Everybody is risk neutral. The current riskless rate is 8%. Each borrower must be allowed to retain a profit of at least $5 in the successful state in order to be induced to apply for a bank loan. You have just learned that 1,000 loan applications have been received after you announced a 45% loan interest rate. You can satisfy only 500. What should be your optimal (profit-maximizing) loan interest rate? Should it be 45% (at which you must ration half the loan applicants) or a higher interest rate at which there is no rationing?

Suppose that T-bills currently have a rate of return of 2%. As- sume that borrowing is possible at the risk free rate. You are risk averse and you are considering constructing a portfolio consisting of T-bills and one of the two risky assets: Stock A or Stock B. You did the following scenario analysis on stocks A and B

Events Bull Market Normal Market Bear Market

Probability Stock Aís return 0.3 50%
0.5 18%
0.2 -20%

Stock Bís return 10%
20%
-15%

1. (a) Compute the expected rate of return and the standard deviation for Stock A and Stock B.

2. (b) Based on the information you have so far, which of the two risky assets, Stock A or Stock B, would you choose to be included in your portfolio with T-bills? Explain.

3. (c) Your friend is considering the same problem but she is more risk averse than you. Should she arrive at a di§erent conclusion than you? Ex- plain.

4. (d) Suppose that you start your portfolio with $1 million and also your portfolio target risk (std dev) is 10% (this is your portfolio from part (b)), how many dollars will you invest in T-bills? What is your port- folioís expected return? Show your work.

2.a. You are running a batch reactor. Each batch takes about four hours to run. You measure the purity of the batch four times in the last hour to ensure that it has stabilized. You want to monitor the results using an X-R chart. The data you have collected for the first 10 batches are given the table below. X1, X2, X3 and X4 are the four samples you pull from the batch in the last hour. Using all the data, find trial control limits for and R charts, construct the chart, and plot the data. (30 pts)

2.b. Is the process in statistical control? Identify out-of-control points. (30 pts)