Detailed answers needed.

1. Let Xi,.... Xio be a random sample from a normal population with mean / and standard deviation 4. Let X be the sample mean and suppose X = 48. Find a 95% confidence interval for A. 2. Each visit to Uris.com has a probability p of resulting in a purchase. Out of a random sample of 500 visits, 15 result in purchases. Use this to find a 95% confidence interval for p. 3. In a random sample of 40 students from the 1999 class at Evian Business School, the av- erage staring salary is $110 thousand and the sample standard deviation is $16 thousand. Find a 95% confidence interval for the population mean. 4. A wireless communication company is considering switching from a per-minute charge to a flat monthly fee for unlimited service. It anticipates that this will result in greater usage and would like to estimate the change in the average number of minutes per month per customer resulting from the new plan. To do this, it offers 850 customers the flat-fee plan for one month and tracks their usage. It also tracks a control group of 700 customers under the old plan. Among the 850 test customers, it finds an average usage of 227 minutes and a sample standard deviation of 135 minutes; in the control group, the corresponding values are 163 and 85. Find a 95% confidence interval for the increase in mean usage. 5. For a t-based confidence interval, what multiplier would you use with a sample size of 15 and a confidence level of 95%? 90%? What confidence level would you get with a multiplier of 2.6247 6. Six months after elections in the Democratic Republic of Urisia, a newspaper reports that two-thirds of French companies with major government contracts have enjoyed increases in the value of their contracts since the election and accuses the government of pro- French policies. Table 3 shows the contract values (in millions of dollars) for the 12 French companies with a major presence in Urisia, before and after the election. Using the t distribution, calculate a 95% confidence interval to assess the difference in average contract size before and after the election. What does this confidence interval suggest? 7. (a) A recent study compared HMOs that provide financial incentives for physicians to reduce referrals with HMOs that do not. The study found that in the first group 12 out of 52 physicians surveyed said they would refer a patient with chest pain upon waking 32 Before After Difference 68.3 71.5 3.2 37.0 32.5 -4.6 121.4 122.1 0.7 113.1 119.8 6.6 117.3 111.2 -6.1 47.7 44.0 -3.6 85.7 94.6 8.8 99.4 103.6 4.2 108.8 129.4 20.6 92.0 74.7 -17.2 63.3 75.1 11.8 33.9 39.2 5.3 Average 2.3 84.8 2.5 Std Dev 31.6 33.8 9.7 Table 3: Data for Problem 6 to a cardiologist; in the second group, 18 out of 45 physicians surveyed said they would make the referral. Find a 95% confidence interval for the difference. (b) Repeat the calculation now assuming the proportions are 120 out of 520 and 180 out of 450. Comment on the difference between this interval and the one in part (a).1. Let X1,..., Xin be a random sample from a population with mean 50 and standard deviation 4. Let X be the sample mean. Find the expected value and standard deviation of X. 2. Each visit to Uris.com has a 2% chance of turning into a purchase. Let p denote the proportion of visits that turn into purchases from a random sample of 100 visits. Find the expected value and standard deviation of p. 3. (a) The starting salary among 1999 graduates of the Evian Business School has a standard deviation of $17,000. If you randomly survey 40 students and average their starting salaries, what is the probability that the average among these students will be greater than the average among all students? b) What is the probability that the average in the sample will exceed the average among all students by more than $5,0007 (c) What is the probability that the average in the sample will differ from the average among all students by more than $5,000? 4. A trader's annual bonus is normally distributed with a mean of $650,000 and standard deviation $125,000. Her bonuses are independent from year to year. Find the probability that her average bonus over a five-year period will be less than $500,000. 5. Fifty-seven percent of students at Calabria Business School support making Introduction to Philanthropy a required course. The school plans to survey 100 students to gauge opinion on this issue. What is the probability that fewer than half of those surveyed will say they support requiring the course? 6. A wireless communication company is considering switching from a per-minute charge to a flat monthly fee for unlimited service. It anticipates that this will result in greater usage and would like to estimate what the average number of minutes per month per customer will be under the new plan. To do this, it offers 1000 customers the flat-fee plan for one month and tracks their usage. From past data, the company knows that the standard deviation of monthly minutes is about 120 minutes; the company expects that this figure will not change much under the new plan. Find the probability that the average among the 1000 test customers will differ from the true mean by more than 5 minutes. 28 7. A company believes that roughly 35%% of consumers rate its brand first in quality. It would like to estimate the proportion more accurately and retains a market research firm to survey a consumers. Using the initial estimate of 35% as a guide, determine how large a must be to ensure that there is only a 5% chance that the estimate from the survey will differ from the true value by more than 3 percentage points.1. Daily demand for widgets is normally distributed with a mean of 100 and a standard deviation of 15. (a) What is the probability that the demand in a day will exceed 125? (b) What is the probability that demand will be less than 757 Less than 70? (c) How many widgets should be stocked to ensure that with 95% probability all demands will be met? 2. The plant manager of a manufacturing facility is concerned about drug use among plant workers and plans to implement random drug testing. One of the tests to be applied measures the level of factor-X in blood samples. Among recent users of cocaine, the level of factor-X is normally distributed with a mean of 10.0 and a standard deviation of 1.3. Among non-users, the level is normally distributed with a mean of 6.75 and a standard deviation of 1.5. The employer plans to send a warning letter to all employees with a factor-X level of r or greater. with r to be determined. (a) Find the value of r that will ensure that 90% of recent cocaine users will be sent a warning letter. (b) If your answer to part (a) is adopted, what proportion of non-users will also be sent warning letters? 3. A bank finds that the one-day increase in the dollar value of its foreign exchange portfolio is normally distributed with a mean of $1.5 million and a standard deviation of $9.7 million. (A negative increase is a loss.) (a) Find the value r such that the probability that the portfolio will lose more than r dollars in one day is 5%. (b) For the r you found in part (a), what is the probability that the portfolio will increase in value by more than r dollars in one day? 4. Mogul Magazine has recently completed an analysis of its customer base. It has deter- mined that 75% of the issues sold each month are subscriptions and the other 25% are sold at newsstands. It has also determined that the ages of its subscribers are normally distributed with a mean of 44.5 and a standard deviation of 7.42 years, whereas the ages of its newsstand customers are normally distributed with a mean of 36.1 and a standard deviation of 8.20 years. 25 (a) Mogul would like to make the following statement to its advertisers: "80% of our subscribers are between the ages of _and _" Your job is to fill in the blanks, choosing a range that is symmetric around the mean. (In other words, the mean age of subscribers should be the midpoint of the range.) (b) What proportion of Mogul's newsstand customers have ages in the range you gave in (a)? (c) What proportion of all of Mogul's customers have ages in the range you gave in (a)?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