1. Research information for all US lotteries and study the summary of state and multi-state lottery odds and prizes. Pick three lotteries and determine the expected value of your winnings in each case. Summarize your findings.
2. Personal Law of Averages . Describe a situation in which you personally have made use of the law of averages, either correctly or incorrectly. Why did you use the law of averages in this situation? Was it helpful?
3. The Gambler’s Fallacy in Life. Describe a situation in which you or someone you know has fallen victim to the gambler’s fallacy. How could the situation have been dealt with correctly?

Should You Play? Suppose someone gives you 10 to 1 odds that you cannot roll a double number (for example, two 1s or two 2s) with the roll of two fair dice. This means you win $10 if you succeed, and you lose $1 if you fail. What is the expected value of this game to you? Should you expect to win or lose the first game? What can you expect if you play 100 times? Explain.

Powerball Lottery. The 17-state Powerball lottery advertises the following prizes and probabilities of winning for a single $1 ticket. Assume the jackpot has a value of $30 million one week. Note that there is more than one way to win some of the monetary prizes (for example, two ways to win $100), so the table gives the probability for each way. What is the expected value of the winnings for a single lottery ticket? If you spend $365 per year on the lottery, how much can you expect to win or lose?

Mean Household Size. It is estimated that 57% of Americans live in households with 1 or 2 people, 32% live in households with 3 or 4 people, and 11% live in house-holds with 5 or more people. Explain how you would find the expected number of people in an American household. How is this related to the mean household size?

House Edge in Roulette. The probability of winning when you bet on a single number in roulette is 1 in 38. A $1 bet yields a net gain of $35 if it is a winner.

1. Suppose that you bet $1 on the single number 23. What is your probability of winning? What is the expected value of this bet to you?
2. Suppose that you bet $1 on each of the numbers 8, 13, and 23. What is your probability of winning? What is the expected value of this bet to you? Remember that you lose your bet on the numbers that do not come up.
3. Compare the results of parts a and b. Does the expected value change with the number of numbers on which you bet?

Behind in Coin Tossing: Can You Catch Up? Suppose that you toss a fair coin 100 times, getting 38 heads and 62 tails, which is 24 more tails than heads.

1. Explain why, on your next toss, the difference in the numbers of heads and tails is as likely to grow to 25 as it is to shrink to 23.
2. Extend your explanation from part a to explain why, if you toss the coin 1,000 more times, the final difference in the numbers of heads and tails is as likely to be larger than 24 as it is to be smaller than 24.
3. Suppose that you continue tossing the coin. Explain why the following statement is true: If you stop at any random time, you always are more likely to have fewer heads than tails, in total.
4. Suppose that you are betting on heads with each coin toss. After the first 100 tosses, you are well on the losing side (having lost the bet 62 times while winning only 38 times). Explain why, if you continue to bet, you will most likely remain on the losing side. How is this answer related to the gambler’s fallacy?

Types of Correlation. Exercises 1–10 list pairs of variables. For each pair, do the following:

1. State the units you would use to measure each of the two variables (for example, pounds, years, or miles per hour). Then state whether you believe the two variables are correlated. If you believe they are correlated, state whether the correlation is positive or negative and strong or weak. Explain your reasoning.
2. Imagine that you actually measured the two variables for a sample of 10 people or items. Draw a rough scatter diagram showing 10 points that you would be likely to find.

2. Height and shoe size

6. Altitude on a mountain hike and surrounding air temperature

8. Weight of car and gas mileage

12. Two-Day Forecast. Figure 7.13 shows a scatter diagram in which the actual high temperature for the day is compared with a forecast made two days in advance.

The data are from the same two weeks as those in Figure 7.6. Estimate the correlation coefficient and discuss what these data imply about weather forecasts. Do you think you would get similar results if you made similar diagrams for other two-week periods? Why or why not?

14. Education and Salary. Consider the following table on mean earnings of men and women in 1997 in terms of education (U.S. Census Bureau, March 1997).

EDUCATION (YEARS EARNINGS, EARNINGS, AFTER HIGH SCHOOL) MEN WOMEN

No HS diploma $17,826 $10,421

0 (HS diploma only) $27,642 $16,161

2 (Associate’s degree) $31,426 $18,933

4 (Bachelor’s degree) $46,702 $28,701

6 (Advanced degree) $74,406 $42,625

Overall $34,705 $20,570

SOURCE: TIME Almanac, 1998.

1. Make scatter diagrams for both men and women, with earnings on the horizontal axis and education on the vertical axis. For purposes of graphing, use 2 years (after high school) for the category "No HS diploma."
2. Briefly characterize each correlation in words and estimate the correlation coefficient for each case.
3. What general conclusions can you reach from the two correlations? Explain

18. Movie Data. Consider the following table on total box office receipts and total attendance for all American films, 1990–1997 (Motion Picture Association of America).

1. Make a scatter diagram for the data.
2. Briefly characterize the correlation in words and estimate the correlation coefficient.
3. How does the fact that the price of movies has increased since 1990 affect these data? If you were a movie executive, what general conclusions from these data would be important to you?

Significance of Correlations. Exercises 22–29 each give a number of data points, n, and a correlation coefficient, r. For each case, state whether the correlation is significant at the 0.05 or 0.01 level, and explain the meaning of your statement.

24. n _ points and r _ .

26. n _ points and r _ .

30. Diamond Significance. The correlation coefficients for the diamond data in Table 7.1 are r _ 0.77 for the correlation between weight and price, and r _ _0.163 for the correlation between color and price. Evaluate the statistical significance of these correlations. (Hint: Use the row in Table 7.3 for 20 data points, even though Table 7.1 actually has 23 data points.)

32. Movie Success. Using the data in Table 7.2, make a scatter diagram for the relationship between production budget and viewer rating of movies. Estimate the correlation coefficient. Based on these data, do you think a large production budget is likely to result in a movie with a high viewer rating? Explain.

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