L04: Homework Answer Key

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Barnaby Carr
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L04: Homework Answer Key Instructions: You are encouraged to collaborate with other students on the homework, but it is important that you do your own work.

1 L04: Homework Answer Key Instructions: You are encouraged to collaborate with other students on the homework, but it is important that you do your own work. Before working with someone else on the assignment, you should attempt each problem on your own. 1. In your own words, list the three rules of probability. A class survey in a statistics class asked students to list which state they were from. The results of the survey are given in the following table. Use this information to answer questions 2 through One of the probabilities is missing from the table. What should that missing probability be, if the table is supposed to be a probability distribution? Which rule of probability helped you answer this question? 38.3% 3. Randomly select a student in the class. According to the table, what is the probability of selecting a student that is from Washington or Oregon? 15.8% 4. Randomly select a student in the class. According to the table, what is the probability of selecting a student that is not from Idaho? Which rule of probability helped you answer this question? 78.1% 5. Randomly select a student in the class. According to the table, what is the probability of selecting a student that is not from Utah or California? 81.4% 6. Randomly select a student in the class. According to the table, what is the probability of selecting a student that is from Texas? It is not possible to answer this question using the data given. There may or may not be students from Texas, but we can t tell how many just from this data.

2 In the finance industry, investors make decisions based on past and present performance of the market. A stock is a share in the ownership of a company. For example, if you were to purchase stock in McDonald's, you would own a small part of that company. If the company does well in the future, the value of your stock would be expected to rise. Then, you could sell your shares in the company for a profit. Stocks are sold in a stock market. We will apply the methods of this lesson to find the mean and standard deviation of the annual change in the value of a stock or bond. Using historical data, a market analyst forecasts that the probability is 0.30 that the value of McDonald s stock will go up by 60% in the next year. The analyst calculates that the probability that the stock will increase by 15% is Some of the analyst s other forecasts are summarized in the table below: 7. Use the information in the table above to determine the probability that McDonald s stock will go down by 10% over the next year By adding two columns to this table, we can compute the mean and standard deviation of the value of McDonald's stock. Use this table to answer questions 8 through Complete the table above. What number should go in the position marked A? Use the table above to find the population mean (i.e. the expected value) for the percentage of growth for McDonald s stock over the next year. Give a number. μ = 19 %

3 10. If you were an executive at McDonald s, how would you interpret the population mean, μ, to a potential investor? Choose the answer that provides the best interpretation of this value. a. The price of McDonald's stock will probably increase over the next year. b. The price of McDonald's stock will probably decrease over the next year. c. The price of McDonald's stock has the potential of varying dramatically over the next year. d. The price of McDonald's stock will probably be fairly consistent over the next year. 11. What number should be entered in the position marked "B"? What is the population variance for the percent change in the price of McDonald s stock over the next year? σ 2 = Compute the population standard deviation of the annual change in the price of McDonald s stock. σ = 29.9 % 14. If you were an executive as McDonald s, how would you interpret the population standard deviation, σ, to a potential investor? Choose the answer that provides the best interpretation of this value. e. The price of McDonald s stock will probably increase over the next year. f. The price of McDonald s stock will probably decrease over the next year. g. The price of McDonald s stock has the potential of varying dramatically over the next year. h. The price of McDonald's stock will probably by fairly consistent over the next year. As you read previously, a stock is a share in the ownership of a company. If the company is perceived as increasing in value, this is usually reflected in an increase in the price per share of the stock. Bonds are used by the U.S. Government to raise money. When a bond is issued, the government promises to pay the purchase price of the bond plus an additional amount as interest at some future date (when the bond matures.) Some investors do not keep the bonds until they mature. These bonds can be bought and sold in the bond market. The price of the bond is determined by the perceived value of the bond at the time of sale. Bonds tend to provide a lower return on the investment, but they tend to be more secure. If an investor purchases a bond and keeps it until it matures, the government promises they will earn a fixed amount of interest. However, when people buy and sell bonds on the bond market, there is the possibility of both large gains and large losses. There is no guarantee that someone who buys and sells either stocks in the stock market or bonds in the bond market will make a profit. To protect against large fluctuations in either the stock market or the bond market, investors diversify their portfolio. They purchase a collection of stocks and bonds that will manage their risk and provide the greatest return possible at that level of risk.

4 Using historical data, the market analyst examined the profitability of investing in a 10-Year bond in the bond market. The probability that the price of a 10-year bond will change by the certain percentages is given below: Use this table to answer questions 15 and Find the population mean for the possible percent change in the price of a 10-year bond over the next year. Round your answer to 2 decimal places and include a minus (-) sign if your answer is negative. µ = % 16. Compute the population standard deviation of the annual change in the price of a 10-year bond. Round your answer to 2 decimal places. σ = % A fair coin is tossed 3 times. Let X be the number of heads observed. The possible values of X are 0,1,2, or 3. The probability of each of these outcomes is given here: X P(x) 1/8 3/8? 1/8 Use this table to answer questions 17 through What is the probability of getting exactly two heads on three tosses? 3/8

5 18. Compute the mean of the distribution of the number of heads observed in three tosses of a fair coin. Round your answer to 1 decimal place. µ = 1.5 % 19. Compute the standard deviation of the distribution of the number of heads observed in three tosses of a fair coin. Round your answer to 3 decimal places. σ = %

Examples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions

Examples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions Random Variables Examples: Random variable a variable (typically represented by x) that takes a numerical value by chance. Number of boys in a randomly selected family with three children. Possible values: