Econ 57
Gary Smith
Fall 2009

Midterm (75 minutes)
No calculators allowed. Just set up your answers; e.g. P = 49/52. If you want extra time, you can buy time at a price of 1 point a minute; for example, if your test is handed in 10 minutes after the scheduled finish time, 10 points will be subtracted from the test score.

1. For each of the following studies, identify the type of graph (boxplot, time series graph, or scatter diagram) that would be the MOST appropriate. You can use more than one graph of each type; for example, two boxplots.

a. Do countries with lots of smokers have lots of lung-cancer deaths?

b. Does the time between eruptions of Old Faithful depend on the duration of the preceding eruption?

c. Is there more variation in annual rainfall in Los Angeles or in New York?

d. Have temperatures in Los Angeles generally increased or decreased over the past 100 years?

e. Could the states that Al Gore won and lost in 2000 have been predicted from how well Bill Clinton did in each state in 1996?

2. In each of the four cases below, which data set has the higher mean? Higher median? Higher standard deviation? Write your answers (A or B) in the table below the cases. (You don’t need to do any calculations.)

a. Case 1

 A 1 2 3 4 5 B 5 4 3 2 1

b. Case 2

 A 1 2 3 4 5 B 1 2 3 4 6

c. Case 3

 A 1 2 3 4 5 B 1 1 3 5 5

d. Case 4

 A 1 2 3 4 5 B 998 999 1,000 1,001 1,002

 Higher Mean Higher Median Higher Standard Deviation Case 1 Case 2 Case 3 Case 4

3. A 1950s study found that married men were in better health than men of the same age who never married or were divorced, suggesting that the healthiest path is for a man to marry and never divorce.

a. Explain why there might be sampling bias in this study.

b. Suppose that marriage is generally bad for a man’s health. Explain how it could still be the case that men who marry and stay married are in better health than: (i) men who don’t marry; and (ii) men who marry and then divorce.

c. What would have to be done to have a controlled experiment?

4. Show how you would use these household taxable income data (in thousands of dollars)

 15,000 23,000 32,000 37,000 48,000 48,000 60,000 80,000 110,000 120,000

to make a histogram, using intervals equal to the federal income tax brackets shown below:

 Taxable Income Bracket Tax \$0 to \$16,000 10% of the amount over \$0 \$16,000 to \$64,000 \$1,600.00 plus 15% of the amount over \$16,000 \$64,000 to \$128,000 \$8,800,00 plus 25% of the amount over \$64,000 \$128,000 to \$208,000 \$24,800.00 plus 28% of the amount over \$128,000 \$208,000 to \$350,000 \$47,200.00 plus 33% of the amount over \$208,000 \$350,000 to no limit \$94,060.00 plus 35% of the amount over \$350,000

5. Use the data in the Question 4 to draw a boxplot summarizing these data.

6. A radio station offered a \$100 prize to a listener who: (a) has a lucky one dollar bill with at least three 9’s among the 8 digits in the dollar’s serial number (for example, 23944199 and 93944199 would be lucky dollars); and (b) is the correct caller, for example, the 47th person to call the radio station during the contest.

a. What is the probability that a randomly chosen dollar bill has at least three 9’s among the 8 digits in the dollar’s serial number? (Assume that the digits are independent and that each digit is equally likely to be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

b. If you have 4 dollar bills, what is the probability that at least one will be a lucky dollar?

7. A routine examination discovers a lump in a female patient’s breast. One out of 100 such lumps turn out to be malignant. If the lump is malignant, there is a 0.80 probability that a mammogram x-ray will identify it as malignant; if the lump is benign, there is a 0.90 probability that the mammogram will identify it as benign. In this particular case, two independent x-rays are taken and each identifies the lump as malignant. What is the revised probability that this lump is malignant?

8. In the gambling game Chuck-A-Luck, a player can bet \$1 on any number from 1 to 6. Three dice are thrown and the payoff depends on the number of times the selected number appears. For example, if you pick the number 2, your payoff is \$4 if all three dice have the number 2. What is the expected value of the payoff?

 Number of dice with number 0 1 2 3 Payoff (dollars) 0 2 3 4

9. Bulgaria has a national lottery in which participants choose 6 of 42 numbers (1 through 42) and win the grand prize if the six numbers they choose match (not necessarily in order) the 6 numbers picked on live television. On September 10, 2009, the six winning numbers (4, 15, 23, 24, 35, 42) were the same six numbers picked on September 6, 2009 (though in a different order). No one had the winning numbers on September 6, but a record 18 people had the winning numbers on September 10. If the game is fair,

a. What is the probability that the 6 numbers picked in any single lottery will be the same 6 numbers picked in the previous lottery (not necessarily in the same order)?

b. What is the expected wait until two consecutive lotteries have the same 6 numbers selected?

10. Suppose that a woman is randomly selected from a population in which heights are normally distributed with a mean of 66 inches and a standard deviation of 2.0 inches and that a male is independently selected from a population in which heights are normally distributed with a mean of 70 inches and a standard deviation of 2.5 inches. For each of the following questions, identify which outcome is more likely (you do NOT have to calculate the probabilities):

a. (i) Her height is more than 70 inches; or (ii) his height is less than 66 inches.

b. (i) Her height is more than 70 inches; or (ii) her height is more than 70 inches and his height is less than 66 inches.

c. (i) Her height is more than 70 inches; or (ii) either her height is more than 70 inches or his height is less than 66 inches.

d. (i) Her height is more than 66 inches and his height is more than 70 inches; or (ii) the sum of their heights is more than 136 inches.