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1. You manage a retail operation from which you sell to both walk-in and telephone cus- tomers. For a particular product, your goal is to set the inventory so that 99% of customers boking or calling for the product find it in stock. Consider the following two scenarios: (i) Days with a lot of walk-in customers are also days with a lot of telephone orders. (ii) Days with a lot of walk-in customers tend to be days with fewer telephone orders and vice-versa. In which scenario would you expect to have to hold more total inventory to meet your service objective? Explain your answer by making reference to the concepts of standard deviation and correlation. 2. You invest $3 thousand in one stock and your spouse invests $2 thousand in another. Over the next year, each dollar invested in your pick will increase by X dollars and each dollar invested in your spouse's will increase by Y dollars; X and Y are random variables with the following properties: o X has a mean of 0.09 and a standard deviation of 0.20. o Y has a mean of 0.12 and a standard deviation of 0.27. o The correlation between X and Y is 0.6. Your individual earnings are 3X thousand, your spouse's individual earnings are 2Y thousand and your family earnings are the sum of two. (a) What is the expected value of your family earnings? (b) What is the standard deviation of your family portfolio earnings? 3. Let X, Y, and Z be random variables. Consider the following statements: (i) if Cou[X, Y] > Cou[Y, Z] then pxy > pyz. (ii) if pxy > pyz then ox > dy. (iii) if pxy > 0 then Cou[X, Y] > 0. Now pick one of the following: (a) all of the above are true (b) (i) and (ii) are true (c) (i) and (iii) are true (d) (ii) and (iii) are true 21 (e) only (iii) is true 4. Suppose you borrow $1000 for one year at a variable interest rate tied to the yield on government bonds. As a result, the total interest you will pay (in dollars) is a random variable X1, having mean 60 and standard deviation 2. You invest the borrowed money. Your earnings on the investment, X2, have mean 85 and standard deviation 8. Suppose the correlation between your earnings and the interest you pay on the loan is 0.3. (a) Your net earnings at the end of the year are Y = X2 - X1. Find the expected value of your net earnings. (b) Find the standard deviation of your net earnings. 5. *** A company knows that it will buy 1 million gallons of jet fuel in 3 months, and wants to hedge against a possible price increase. The company chooses to hedge by buying futures contracts on heating oil. Suppose the standard deviation of changes in the price per gallon of jet fuel over 3 months is 0.032, the standard deviation of changes in the futures price per gallon of heating oil is 0.040, and the correlation between the two is 0.8. Also, each heating oil futures contract is for 42,000 gallons. (a) What is the standard deviation of the company's unhedged exposure? (Think of the company as holding -1,000, 000 gallons of jet fuel, because of its anticipated purchase.) (b) A simple gallon-for-gallon hedging rule would suggest that the company should buy 1, 000,000/42,000 = 23.8 = 24 contracts. What is the standard deviation of the com- pany's exposure under this strategy? (Hint: Let X = change in price of jet fuel and Y = change in futures price (per gallon) of heating oil. Write the exposure in terms of X and Y. ) (c) Find the number of contract that minimizes the standard deviation of the company's exposure.7-1 1. Data on pull-off force (pounds) for connectors used in an automobile engine application are as follows: 79.3, 75.1, 78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7. 75.9. 72.9, 73.8. 74.2. 78.1, 75.4, 76.3, 75.3, 76.2, 74.9, 78.0. 75.1, 76.8. (a) Calculate a point estimate of the mean pull-off force of all connectors in the population. State which estimator you used and why. (b) Calculate a point estimate of the pull-off force value that separates the weakest 50% of the connectors in the popu- lation from the strongest 50% (c) Calculate point estimates of the population variance and the population standard deviation. (d) Calculate the standard error of the point estimate found in part (a). Provide an interpretation of the standard error. (e) Calculate a point estimate of the proportion of all connec- tors in the population whose pull-off force is less than 73 pounds.(a) Construct a normal probability plot of each of the data sets. Based on these plots, is it reasonable to assume that they are each drawn from a normal population? (b) Suppose that it was determined that the lowest observa- tion of the Mercedes data was erroneously recorded and should be 24.6. Furthermore, the lowest observation of the Volkswagen data was also mistaken and should be 39.6. Again construct normal probability plots of each of the data sets with the corrected values. Based on these new plots, is it reasonable to assume that they are each drawn from a normal population? (c) Compare your answers from parts (a) and (b) and com- ment on the effect of these mistaken observations on the normality assumption. (d) Using the corrected data from part (b) and a 95% confi- dence interval, is there evidence to support the claim that the variability in mileage performance is greater for a Volkswagen than for a Mercedes?10-73. The Salk polio vaccine experiment in 1954 focused on the effectiveness of the vaccine in combatting paralytic polio. Because it was felt that without a control group of children there would be no sound basis for evaluating the efficacy of the Salk vaccine, the vaccine was administered to one group, and a placebo (visually identical to the vaccine but known to have no effect) was administered to a second group. For ethical reasons, and because it was suspected that knowledge of vaccine administration would affect subse- quent diagnoses, the experiment was conducted in a double- blind fashion. That is, neither the subjects nor the administrators knew who received the vaccine and who received the placebo. The actual data for this experiment are as follows: Placebo group: n = 201,299: 110 cases of polio observed Vaccine group: n = 200,745: 33 cases of polio observed (a) Use a hypothesis-testing procedure to determine if the proportion of children in the two groups who contracted paralytic polio is statistically different. Use a probability of a type I error equal to 0.05. (b) Repeat part (a) using a probability of a type I error equal to 0.01. (c) Compare your conclusions from parts (a) and (b) and ex- plain why they are the same or different.10-17. The diameter of steel rods manufactured on two dif- ferent extrusion machines is being investigated. Two random samples of sizes n, = 15 and n, = 17 are selected, and the sample means and sample variances are ], = 8.73, s = 0.35, X, = 8.68, and $ = 0.40. respectively. Assume that of = of, and that the data are drawn from a normal distribution. (a) Is there evidence to support the claim that the two ma- chines produce rods with different mean diameters? Use o = 0.05 in arriving at this conclusion. (b) Find the P-value for the r-statistic you calculated in part (a). (c) Construct a 95% confidence interval for the difference in mean rod diameter. Interpret this interval