Advanced Algebra
Chapter 11 Answers Advanced Techniques for Chance and Data
In the following investigations, you'll use data to calculate probabilities relating to U.S. governors, U.S. senators, and voting patterns in the 1996 Presidential election.
Part B, The Probability of A or B
1. Use data on U.S. governors found at the Governor
Web site. to find the probability that a randomly chosen state
governor meets the given conditions. (Note: The list includes
five non-state governors, the governors of American Samoa, Guam,
Northern Mariana Islands, Puerto Rico, and the Virgin Islands.)
[All answers correct as of 7 / 98.]
a. The governor is a Republican.
[16 / 25 = 64%]
b. The governor is not a Democrat.
[33 / 50 = 66% (One governor is an Independent.)]
c. The governor is the governor of your state.
[1/50 = 2%]
d. The governor served at least one previous term.
[23 / 50 = 46%]
e. The governor is a woman. Use data from the Women Governors Web site to answer this question.
[3 / 50 = 6%]
f. The governor is a woman or is serving a regular term of 2 years.
[3 / 50+ 2 / 50 1 / 50 = 2 / 25 = 8%]
g. The governor is a lawyer or an Independent. Use data from the Search Governors Web site to answer this question. (Note that the list of lawyers includes the governor of American Samoa.)
[24 / 50 + 1 / 50 1 / 50 = 24 / 50 or 48%]h. Use data on U.S. governors to give an example of an event and the complement of the event. Show that the sum of the probabilities of the events equals 1.
[Answers will vary; event:
The governor is male (P = 47 / 50 );
complement: The governor is not male
(P = 3 / 50 ); 47 / 50 + 3 / 50 = 50 / 50 = 1]
Part B, Independent and Dependent
Events
2. Use the Internet to answer the questions below.
a. What is the probability that a randomly chosen U.S. voter who turned out for the 1996 Presidential election was from Connecticut? Use the data from the Voter Registration Web site to answer this question.
[ 1,392,614 / 96,456,345 ≈ 1.44%]b. What is the probability that a randomly chosen voter in the 1996 election voted for Green Party candidate Ralph Nader? Use the data from the Popular Vote Web site to answer this question.
[0.71%]c. Based on your answers to a and b, what is the probability that a randomly chosen voter in the 1996 election was from Connecticut and voted for Ralph Nader?
[0.0102%]d. Based on the actual Connecticut vote, what is the probability that a randomly chosen voter in the 1996 election was from Connecticut and voted for Ralph Nader? Refer to the Presidential Election Results Web site.
[24,321 / 96,456,345 ≈ 0.0252%]e. What conclusion can you draw from these results?
[Nader's showing in Connecticut was considerably stronger than his average showing around the country. In Connecticut, he polled
0.0252 / 0.0102 ≈ 2.5 times his U.S. average.]
Part C, Conditional Probability
3. Using the Internet, answer the following questions.
a. What is the probability that a randomly chosen U.S. Senator whose term expires in 2001 is a Democrat? Use the data from the Class Membership Web site to answer this question.
[14 / 33 ≈ 42.4%]b. What is the probability that a randomly chosen senator from Ohio is a Republican? Use the data from the Directory of Senators by State Web site to answer this question.
[ 1 /2 = 50%]c. What is the probability that a randomly chosen senator is a Democrat who serves on the Committee on Agriculture, Nutrition and Forestry? Use the data from the Committe and Subcommitte Membership Web site to answer this question.
[ 8 / 100 = 8% or (2 / 25 = 8%)]
d. What percent of the senators in the above group (3c) are from South Dakota?
[25%]e. Show how you can use your answers to (3c) and (3d) to find the probability that a randomly chosen senator is a South Dakota Democrat who serves on the Committee on Agriculture, Nutrition and Forestry.
[Find the product of the answers. 0.08 x 0.25 = 0.02 or 2%]
Sampling methods have improved greatly since the 1936 Presidential race, when polls predicted that Alf Landon would defeat Franklin D. Roosevelt, and the 1948 race, when Thomas Dewey was forecasted as the winner over Harry Truman.
Part A, Biased and Unbiased Sampling
1. Use the Internet to answer the following questions.
a. Use the Gallop Organization Web site to make a histogram displaying the results of the Gallup Presidential Election Poll for October 30-31, 1996.
b. Assuming that the sample used for the poll was representative, find the probability that a randomly chosen person was undecided or supported the Perot-Choate ticket on October 30-31, 1996.
[14%]c. Compare the results of the 1996 election predicted by the final Gallup poll with the actual results at the Popular Vote Summary Web site. Was the poll accurate?
[Gallup: 52%, 41%, 7%, 0%; actual: 49.24%, 40.71%, 8.40%, 1.65% ("other"). Answers will vary. The poll was quite accurate, predicting Dole-Kemp to the nearest percent and Perot-Choate within 1 percent. In the poll, no allowance was made for 4th-party candidates, who received 1.65% of the vote. This may explain the slightly inaccurate prediction of 52% for Clinton-Gore, who actually received 49.24% of the vote.]d. Read about the methodology used by the Gallup Organization in its presidential polling at the Gallup Organization Methodology Web site. Give two examples of how the pollsters attempted to eliminate bias from their sample.
[Answers will vary. Pollsters used random numbers to choose phone numbers of people who were called, and they weighted their results to correct for the possibility that they may have called the same person twice, because that person had more than one phone line.]
Part C, Making Connections
2. Use the following Web sites to answer these questions.
a. Make a histogram of the heights of U.S. Presidents, grouping data in 1-in. intervals at the Lists of U.S. Presidents Web site.
[Art: bars measure 52%, 34%, 10%, 4% respectively]b. Do you think a normal distribution is a reasonable model for this data? Why or why not?
[Answers will vary. Students should point out that the shape resembles a normal curve except for the tall bars at 72 in. and 74 in. These distortions prevent a normal distribution from reasonably modeling the data. Students should understand that normal curves usually become apparent only after large amounts of data are graphed, far more than are available here.]c. Choose a set of Presidential data that in raw form appear to be normally distributed. Make a histogram of the data and tell whether it confirms or refutes your conjecture.
[Choices of data will vary. Three possible histograms are shown below. Each bears a slight resemblance to a normal curve and could take on the classic normal curve shape after many more people have served as President.
Length of Inaugural Speech. Intervals of 500, omitting data point 8444.
Share of Popular Vote. Intervals of 2.5%.
Age at Marriage. Intervals of 2 years.
]
d. Give an example of a set of Presidential data that are clearly not normally distributed. Explain why they are not.
[Answers will vary. "Number of children" is not normally distributed. The most common data (0-2 children) are bunched on the left. Data become less and less common as we move to the right, because the more children in a family, the less common families of that size are.]
