12.5 Homework Solutions

1. 12.5 Homework Solutions 27. (a) 28. (b) 29. (d) 30. (e) 53. Positive Correlation, Weak 54. Negative Correlation, Moderate 55. No Correlation 56. Negative Correlation, Weak • The sign of the correlation shows whether the correlation is pos/neg; the closer to 1, the stronger the correlation. Math 132: Foundations of Mathematics

2. Math 132:Foundations of Mathematics Amy Lewis Math Specialist IU1 Center for STEM Education Math 132: Foundations of Mathematics

3. 14.1 Voting Methods • Understand and use preference tables. • Use the following methods to determine an election’s winner: • Plurality • Borda count • Plurality-with-elimination • Pairwise comparison Math 132: Foundations of Mathematics

4. Preference Tables • Preference ballots: ballots in which a voter is asked to rank all of the candidates in order of preference. • Preference table: a table that shows how often each particular outcome occurred. • Refer to the preference table on page 773. Math 132: Foundations of Mathematics

5. Preference Tables • How many students voted in the election? • How many students selected the candidates in this order: B, S, A, C? • How many students selected Samir (S) as their first choice for student body president? Math 132: Foundations of Mathematics

6. Popular Voting Methods • The plurality method • The Borda count method • The plurality-with-elimination method • The pairwise comparison method Math 132: Foundations of Mathematics

7. The Plurality Method • The candidate (or candidates, if there is more than one) with the most first-place votes is the winner. • A plurality occurs when no single candidate receives a majority of first-place votes (more than 50% of the votes). Math 132: Foundations of Mathematics

8. The Plurality Method • Four candidates are running for mayor of Smallville: Antonio (A), Bob (B), Carmen (C), and Donna (D). The voters were asked to rank all the candidates in order of preference. • Who is declared the winner using the plurality method? Math 132: Foundations of Mathematics

9. The Borda Count Method • Developed by the French mathematical and naval captain Jean-Charles de Borda. • Assigns points to each candidate based on how they are ranked by the voters: • Last-place: 1 pt. • Second-to-last-place: 2 pts. • Third-from-last-place: 3 pts. • Etc. • The points are totaled for each candidate separately. • The candidate with the most points is the winner. Math 132: Foundations of Mathematics

10. The Borda Count Method • Who is declared the winner using the Borda Count method? Math 132: Foundations of Mathematics

11. The Borda Count Method • A gets 520 + 120 + 200 + 300 = 1140 points • B gets 390 + 360 + 300 + 450 = 1500 points • C gets 260 + 240 + 100 + 600 = 1200 points • D gets 130 + 480 + 400 + 150 = 1160 points • Bob wins! Math 132: Foundations of Mathematics

12. The Plurality-with-Elimination Method • The candidate with the majority of first-place votes wins. • If no candidate receives a majority of first-place votes, • eliminate the candidate with the fewest first-place votes. • Move the candidates in each column below the eliminated candidate up one place. • The candidate with the majority of first-place votes in the new preference table wins. • Repeat the process until a candidate receives a majority. Math 132: Foundations of Mathematics

13. The Plurality-with-Elimination Method • Does any candidate have the majority? • Who do we eliminate? • What does the new preference table look like? Math 132: Foundations of Mathematics

14. The Plurality-with-Elimination Method • Does any candidate have the majority now? • Who do we eliminate? • What does the new preference table look like? Math 132: Foundations of Mathematics

15. The Plurality-with-Elimination Method • Does any candidate have the majority now? • Who wins? • Carmen wins! Math 132: Foundations of Mathematics

16. Pairwise Comparison Method • The preference table is used to make a series of comparisons in which each candidate is compared to each of the other candidates. • For each pair of candidates, X and Y, use the table to determine how many voters prefer X to Y and vice versa. • If a majority prefer X to Y, then X receives 1 point. If a majority prefer Y to X, then Y receives 1 point. If the candidates tie, then each receives ½ point. • After all comparisons have been made, the candidate receiving the most points is the winner. Math 132: Foundations of Mathematics

17. Pairwise Comparison Method • How many comparisons do we need to make? • Antonio vs. Bob • Antonio vs. Carmen • Antonio vs. Donna • Bob vs. Carmen • Bob vs. Donna • Carmen vs. Donna Math 132: Foundations of Mathematics

18. Pairwise Comparison Method Antonio vs. Bob • Bob gets 1 point. Math 132: Foundations of Mathematics

19. Pairwise Comparison Method Antonio vs. Carmen • Antonio gets ½ pt. • Carmen gets ½ pt. Math 132: Foundations of Mathematics

20. Pairwise Comparison Method Antonio vs. Donna • Antonio gets ½ pt. • Donna gets ½ pt. Math 132: Foundations of Mathematics

21. Pairwise Comparison Method Bob vs. Carmen • Bob gets 1 point Math 132: Foundations of Mathematics

22. Pairwise Comparison Method Bob vs. Donna • Bob gets ½ pt. • Donna gets ½ pt. Math 132: Foundations of Mathematics

23. Pairwise Comparison Method Carmen vs. Donna • Carmen gets ½ pt. • Donna gets ½ pt. Math 132: Foundations of Mathematics

24. Pairwise Comparison Method • Who wins? • Antonio: 1 point • Bob: 2½ points • Carmen: 1½ points • Donna: 1 point • Bob wins! Again! Math 132: Foundations of Mathematics

25. Who were our winners? • Plurality: Donna • Borda count: Bob • Plurality-with-elimination: Carmen • Pairwise comparison: Bob • Who should be mayor of Smallville?!? Math 132: Foundations of Mathematics

26. Homework Page 782: #7 Apply all 4 voting methods to determine the kind of play the theater society will perform next semester. Next Session: Thursday, May 20 Math 132: Foundations of Mathematics