Ch12 Exercises

4/20/2008

12 Exercises Need to add kurtosis, skewness…

Mix and Match

In Questions 1-10, the X denotes a normally distributed random variable: X ~ N(µ,σ). A googol, the namesake of Google, is 10100.

1. Symbol for the mean value of X a. ½

2. Symbol for the variance of X b. P(Z < 1)

3. Probability of a normal r.v. being less than its mean c. 2/3

4. Probability of a normal r.v. being less than 1 SD above its mean d. 0.05

5. Standard deviation of a standardized normal r.v. e. 1/googol

6. Probability that a z-score based on X is less than 1 in magnitude. f. µ

7. Proportion of a normal distribution that is more than 20σ from µ . g. 0

8. Numerical difference between P(Z < -x) and P(Z > x) h. 1

9. Distribution formed by the calculation µ + σ Z i.

€

σ 2

10. Probability that Z > 1.96 plus the probability that Z < –1 j. N(µ, σ)

True/False If you believe that a statement is false, briefly explain why you think it is false.Questions 11-16.

The current age (in years) of 400 clerical employees at an insurance claims processing center is normally distributed with mean 38 and SD 6.

11. More employees at the processing center are older than 44 than between 38 and 44.

12. None of the employees at this center is younger than 25.

13. If the company holding the records were to convert these ages from years to days then the age in days would also be normally distributed.

14. If none of these employees leaves the firm and no new hires are made, then the distribution a year from now will be normal with mean 39 and SD 6.

15. A training program for employees under the age of 30 at the center would be expected to attract about 36 employees.

16. If the ages of these employees were mixed with those from other divisions (Management, Sales, Accounting, Information Systems), then the distribution would also be normal.

Questions 17-22. The number of packages handled by a freight carrier daily is normally distributed. On average, 8,000 packages are shipped each day, with standard deviation 600. Assume the package counts are independent from one day to the next.

4/20/2008 12 Exercises

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17. The total number of packages shipped over 5 days is normally distributed.

18. The difference between the numbers of packages shipped on any two consecutive days is normally distributed.

19. The difference between the number of packages shipped today and the number shipped tomorrow is zero.

20. .If each shipped package earns the carrier $10, then the amount earned per day is normally distributed.

21. The probability that more packages are handled tomorrow than today is ½.

22. To verify the use of a normal model, the shipper should use the normal quantile of the amounts shipped on a series of days.

Think About It

23. How large can a single observation from a normal distribution be?

24. If used to model data that have limits (such as the age of a customer using a credit card or amount spent by a customer), what problem does a normal model have?

25. If X1 ~ N(µ, σ) and X2 ~ N(µ, σ) are iid normal random variables, then what’s the difference between 2 X1 and X1 + X2? They both have normal distributions.

26. If X1 ~ N(µ, σ) and X2 ~ N(µ, σ) are iid, then what is the distribution of (X1 – X2)/(√2σ)?

27. Look at these normal quantile plots. Which one of the following normal quantile plots indicates that the data … a. Are nearly normal? b. Have a bimodal distribution? (One way to recognize a bimodal shape is a “gap” in the spacing of adjacent data values.) c. Are skewed? d. Have outliers to both sides of the center?

A B C D

.01 .05.10 .25 .50 .75 .90.95 .99

.01 .05.10 .25 .50 .75 .90.95 .99.01 .05.10 .25 .50 .75 .90.95 .99

.01 .05.10 .25 .50 .75 .90.95 .99

28. Match the histogram in the column on the left to the normal quantile plot shown on the

right.


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a

.01 .05.10 .25 .50 .75 .90.95 .99

A

b

.01 .05.10 .25 .50 .75 .90.95 .99

B

c

.01 .05.10 .25 .50 .75 .90.95 .99

C

d

.01 .05.10 .25 .50 .75 .90.95 .99

D

29. One of these pairs of histogram with normal quantile plots shows data that has been rounded. The other shows the same data without rounding. The rounding in the example is similar to the rounding in SAT scores. (a) Which is which? How can you tell? (b) Why do the histograms appear so similar if the data are different?


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A B

.01 .05.10 .25 .50 .75 .90.95 .99

.01 .05.10 .25 .50 .75 .90.95 .99

30. The normal quantile plot of daily stock returns on General Motors during 1992-1993 (507

trading days) shows an anomaly, a flat spot at zero. (a) What’s happened? (b) Why does the anomaly not appear in the histogram?

-0.07-0.06-0.05-0.04-0.03-0.02-0.01

00.010.020.030.040.050.060.070.08

25 50 75 125Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3

Normal Quantile Plot

.25 .50 .75

31. The amount of electricity supplied by a utility to residences in a community during the

summer depends on the local weather as well as the habits of the people in the community. (a) Would a normal model be reasonable to describe the amount electricity used from one day to the next? Explain your answer briefly. (b) Would it be reasonable to treat a sequence of daily electricity consumption as iid normal random variables? Again, explain your answer briefly.

32. Dress shoes of a specific style for women come in sizes ranging from 4 to 12. A retailer believes that this range of sizes is adequate for all but 5% of its female customers. (a) What normal model for sizes is consistent with the opinion of the retailer? (b) If the retailer uses a normal model, then should it order more shoes of sizes 4 & 5 or more shoes of sizes 11 & 12? (c) Can a normal model possibly be the right model for the proportion of shoe sizes of its female customers?

33. The weekly salary paid to employees of a small company that supplies part-time laborers averages $700 with standard deviation = $400.


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(a) If the weekly salaries are normally distributed, estimate the fraction of employees that make more than $300 per week.

(b) If every employee receives a year-end bonus that adds $100 to the paycheck in the final week, how does this change the normal model for that week?

(c) If every employee receives a 5% salary increase for the next year, how does the normal model change?

(d) If the lowest salary is $300 and the median salary is $500, does a normal model appear appropriate?

34. A specialty foods company mails out “gourmet steaks” to customers willing to pay a gourmet price. The steaks vary in size, with a mean uncooked weight of 1.2 pounds and standard deviation of 0.10 pounds. Use a normal model for the weight of a steak. (a) Estimate the proportion of steaks that weigh less than 1 pound. (b) If a standard order has 5 steaks, then what model can be used to describe the total weight

of the steaks in a standard order? (Do you need to make any assumptions?) (c) If the weights are converted to ounces (16 ounces = 1 pound), then what normal model

would be appropriate for the weight of an individual steak? (d) The lower quartile, median, and upper quartile of the weights of the steaks are 1.125

pounds, 1.2 pounds, and 1.275 pounds. Do these summaries suggest that the use of a normal model is appropriate?

35. A contractor built 30 similar homes in a suburban development. The homes have comparable size and amenities, but each has been sold with features that customize the appearance, landscape, and interior. The contractor expects the homes to sell for about $450,000. He expects that 1/3 of the homes will sell either for less than $400,000 or more than $500,000. (a) Would a normal model be appropriate to describe the distribution of sale prices? (b) What data would help you decide if a normal model is appropriate? (You cannot use the prices of these 30 homes; the model is to describe the prices of as-yet unsold homes.) (c) What normal model has properties that are consistent with the intuition of the contractor?

36. An accounting firm assists small businesses file annual tax forms. It assigns each new client to a CPA at the firm who specializes in companies of that type and size. For instance, one CPA specializes in boutique clothing retailers with annual sales of about $2 million. To speed the process, each business submits a preliminary tax form to the accounting firm. (a) Would a normal model be useful to describe the total size of adjustments when a CPA reviews the preliminary tax forms? (For example, suppose the preliminary form claims that the business owes taxes of $40,000. If form completed by the CPA says the tax obligation is $35,000, then the adjustment is –$5,000.) (b) What data would help you decide if a normal model is appropriate? (You cannot data from the current year; that data is not yet available.) (c) If the average adjustment obtained by the CPA who specializes in clothing retailers is –$7,000 (i.e., $7,000 less than indicated on the preliminary form), then what SD implies that all but a few business end up with lower taxes after the work of these accountants? (Assume a normal model for this question.) (d) Would a normal model be useful to describe the total size of adjustments for all of the CPAs at this accounting firm?


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You Do It

37. Find these probabilities for a standard normal random variable Z. Be sure to draw a picture to check your calculations. Use the normal table in the front of the book or software. (a) P(Z < 1.5) (b) P(Z > -1) (c) P(|Z| < 1.2) (d) P(|Z| > 0.5) (e) P(-1 ≤ Z ≤ 1.5)

38. Find these probabilities for a standard normal random variable Z. Be sure to draw a picture for each to check your calculations. Use the normal table in the front of the book or software. (a) P(Z ≥ 0.3) (b) P(Z < -2.2) (c) P(-0.7 ≤ Z < 0.7) (d) P(|Z| > 1.5) (e) P(0.3 ≤ Z ≤ 2.2)

39. Using the normal table in the front of the book or software, find the value of z that makes the following probabilities true. You might find it helpful to draw a picture to check your answers. (a) P(Z > z) = 0.20 (b) P(Z ≤ z) = 0.50 (c) P(-z ≤ Z ≤ z) = 0.50 (d) P(|Z| > z) = 0.01 (e) P(|Z| < z) = 0.90

40. Using the normal table in the front of the book or software, find the value of z that makes the following probabilities hold. Again, you might find it helpful to draw a picture to check your answers. (a) P(Z < z) = 0.25 (b) P(Z ≥ z) = 0.60 (c) P(-z ≤ Z ≤ z) = 0.30 (d) P(|Z| > z) = 0.005 (e) P(|Z| < z) = 0.99

41. (a) Find the value at risk (VaR) for an investment of $100,000 at 2%. (That is, find out how low the value of this investment could be if we rule out the worst 2% of outcomes.) The investment is expected to grow during the year by 8% with SD 20%. Assume a normal model for the change in value. (b) To reduce the value at risk to $20,000, how much more expected growth would be necessary? Assume that the SD of the growth remains 20%. (c) If the value at risk of an investment is $20,000 for a one-year holding period, does that mean that the value at risk for a two-year holding period is $40,000?

42. (a) Find the value at risk for an investment of $500,000 at 1%. (That is, find out how low the value of this investment could be if we rule out the worst 1% of outcomes.) The investment is expected to grow during the year by 10% with SD 35%. Assume a normal model for the change in value. (b) How much smaller would the annual SD of the investment need to become in order to


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reduce the value at risk to $200,000? Assume that the expected growth remains 10%. (c) If this performance is expected to continue for two years, what is the value at risk (at 1%) if the investment is held for two years rather than one? Use the features of the investment in “a.”

43. An insurance company found that 2.5% of male drivers between the ages of 18 to 25 are involved in serious accidents annually. To simplify the analysis, assume that every such accident costs the insurance company $65,000 and that a driver can only have one of these accidents in a year. (a) If the company charges $2,500 for such coverage, what is the chance that it loses money on a single policy? (b) Suppose that the company writes 1,000 such policies to a collection of drivers. What is the probability that the company makes a profit on these policies? Assume that the drivers don’t run into each other and behave independently. (c) Does the difference between “a” and “b” explain how insurance companies stay in business? Large auto insurers are certainly profitable. A recent report, for example, claims that Allstate pays out less than $0.50 in accident claims for every dollar collected in premiums. (Business Week, 5/1/2006)

44. A tire manufacturer warranties its tires to last at least 20,000 miles or “you get a new set of tires.” In its experience, a set of these tires lasts on average 26,000 miles with SD 5,000 miles. Assume that the wear is normally distributed. The manufacturer profits $200 on each sold set, and replacing a set costs it $400. (a) What is the probability that a set of tires wears out before 20,000 miles? (b) What is the probability that the manufacturer turns a profit on selling a set to one customer? (c) If the manufacturer sells 500 sets of tires, what is the probability that it earns a profit after paying for any replacements. Assume that the purchases are made around the country and that the drivers experience independent amounts of wear.

45. A hurricane bond pays the holder a face amount, say $1 million, if a hurricane causes major damage in the US. Suppose that the chance for such a storm is 5% per year. (a) If a financial firm sells these bonds for $60,000, what is the chance that the firm loses money if it only sells one of these? (b) If the firm sells 1,000 of these policies, each for $60,000, what is the probability that it loses money? (c) How does the difference between “a” and “b” compare to the situation of a life insurance company that writes coverage to numerous patients that live or die independently of one another?

46. Hedge Funds These data are returns of 533 hedge funds reported for the month of December 2005. The returns are computed as the change in value of assets managed by the fund during the month divided by the value of the assets at the start of the month. (a) Describe the histogram and boxplot of the returns. Is the histogram symmetric and unimodal? Would you say that this histogram is bell-shaped? (b) Are there any outliers? Which fund had the highest return? The lowest? (c) What normal model describes the distribution of returns of these hedge funds?


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(d) Is the normal model in part “c” a good description of the returns? Explain how you have decided whether it’s a good match or not.

47. Pharma Promotion These data give the market share (0 to 100%) of a brand name pharmaceutical product in 224 metropolitan locations. The market share indicates the percentage of all prescriptions for this type of medication that are written for this brand. (a) Describe the histogram and boxplot of the market shares. Is the histogram symmetric and unimodal? Would you say that this histogram is bell-shaped? (b) Are there any outliers? In what location does this brand have the highest market share? The lowest? (c) The market share can only be as large as 100 and cannot be negative. Since a normal model allows for any value, is it possible for a normal model to describe these shares? Would a normal model be reasonable if the average share were closer to 0 or 100%? (d) What normal model describes the distribution of returns of these shares? (e) Is the normal model in part “d” a good description of these shares? Explain how you have decided whether it’s a good match or not.


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4M Normality of Stock Returns

Percentage changes (or returns) on the stock market follow a normal distribution – if we don’t reach too far into the tails of the distribution. Are returns on stocks in individual companies also roughly normally distributed? This example uses monthly returns on McDonald’s stock from 1980 through the end of 2005 (312 months).

Motivation a) If the returns on stock in McDonald’s during this period are normally distributed, then

how can the performance of this investment be summarized? b) Should the analysis be performed using the stock price, the returns on the stock, or the

percentage changes? Does it matter?

Method c) Before summarizing the data with a histogram, what assumption needs to be checked

first? d) What plot should be used to check for normality, if the data can be summarized with a

histogram?

Mechanics e) If the data are summarized with a histogram, what features are lost and are these too

important to conceal? f) Does a normal model offer a good description of the percentage changes in the price of

this stock? g) Using a normal model, estimate the chance that the price of the stock will increase by

10% or more in at least one month during the next 12 months. h) Count the historical data rather than use the normal model to estimate the probability of

the price of the stock going up by 10% or more in a coming month. Does the estimate differ from that given by a normal model?

Message i) Describe these data using a normal model, pointing out strengths and any important

weaknesses or limitations.


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4M Normality and Transformation

We usually do not think of the distribution of income as being normally distributed. Most histograms show the data to be skewed, with a long right tail reaching out toward Bill Gates. This sort of skewness is so common, but normality so useful, that the lognormal model has become popular. The lognormal model says that the logarithm of the data follow a normal distribution. (It does not matter which log you use because one log is just a constant multiple of another.)

For this exercise, we’ll use a sample of incomes from the 2000 Census. These gives the total household income for a sample of 258 residences in San Antonio, Texas.

Motivation a) What advantage would there be in using a Normal model for the logs rather than a

model that described the skewness directly? b) If poverty in this area is defined as having a household income less than $20,000, how can

you use a lognormal model to find the percentage of households in poverty?

Method c) These data are reported to be a sample of households in the San Antonio area. If the data

are from one cluster in a cluster sample, would that cause problems in using this data? d) How do you plan to check whether the lognormal model is appropriate for these

incomes?

Mechanics e) Does a normal model offer a good description of the household incomes? Explain. f) Does a normal model offer a good description of the log10 of household incomes?

Explain. g) Using the lognormal model with parameters set to match this sample, find the

probability of finding a household with income less than $20,000. h) Is the lognormal model suitable for determining this probability?

Message i) Describe these data using a lognormal model, pointing out strengths and any important

weaknesses or limitations.

Ch12 Exercises

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