12.5 homework solutions
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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 - PowerPoint PPT PresentationTRANSCRIPT
Math 132: Foundations of MathematicsMay 19, 2010
12.5 Homework Solutions27. (a)28. (b)29. (d)30. (e)53. Positive Correlation, Weak54. Negative Correlation, Moderate55. No Correlation56. Negative Correlation, Weak57. The sign of the correlation shows whether the
correlation is pos/neg; the closer to 1, the stronger the correlation.
Math 132: Foundations of MathematicsMay 19, 2010
Math 132:Foundations of Mathematics
Amy LewisMath Specialist
IU1 Center for STEM Education
Math 132: Foundations of MathematicsMay 19, 2010
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
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.
May 19, 2010
Math 132: Foundations of Mathematics
Preference TablesPreference Table for the Election of Student Body President
Number of Votes 2100 1305 765 40
First Choice S A S B
Second Choice A S A S
Third Choice B B C A
Fourth Choice C C B C
May 19, 2010
• 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
Popular Voting Methods
• The plurality method• The Borda count method• The plurality-with-elimination method• The pairwise comparison method
May 19, 2010
Math 132: Foundations of Mathematics
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).
May 19, 2010
Math 132: Foundations of Mathematics
The Plurality Method
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
May 19, 2010
• Who is declared the winner using 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.
Math 132: Foundations of Mathematics
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.May 19, 2010
Math 132: Foundations of Mathematics
The Borda Count MethodPreference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
May 19, 2010
• Who is declared the winner using the Borda Count method?
Math 132: Foundations of Mathematics
The Borda Count MethodPreference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice: 4 pts. A: 130*4 = 520
D: 120*4 = 480
D: 100*4 = 400
C: 150*4 = 600
Second Choice: 3 pts. B: 130*3 = 390
B: 120*3 = 360
B: 100*3 = 300
B: 150*3 = 450
Third Choice: 2 pts. C: 130*2 = 260
C: 120*2 = 240
A: 100*2 = 200
A: 150*2 = 300
Fourth Choice: 1 pt. D: 130*1 = 130
A: 120*1 =120
C: 100*1 = 100
D: 150*1 = 150
May 19, 2010
• 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
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.
May 19, 2010
Math 132: Foundations of Mathematics
The Plurality-with-Elimination Method
• Does any candidate have the majority?• Who do we eliminate?• What does the new preference table look like?
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Math 132: Foundations of Mathematics
The Plurality-with-Elimination Method
• Does any candidate have the majority now?• Who do we eliminate?• What does the new preference table look like?
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice B D D C
Second Choice C B B B
Third Choice D C C D
Math 132: Foundations of Mathematics
The Plurality-with-Elimination Method
• Does any candidate have the majority now?• Who wins?• Carmen wins!May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice C D D C
Second Choice D C C D
Math 132: Foundations of Mathematics
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.
May 19, 2010
Math 132: Foundations of Mathematics
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
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Bob gets 1 point.
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Antonio vs. Bob
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Antonio gets ½ pt.• Carmen gets ½ pt.
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Antonio vs. Carmen
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Antonio gets ½ pt.• Donna gets ½ pt.
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Antonio vs. Donna
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Bob gets 1 point
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Bob vs. Carmen
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Bob gets ½ pt.• Donna gets ½ pt.
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Bob vs. Donna
Math 132: Foundations of Mathematics
Pairwise Comparison Method
• Carmen gets ½ pt.• Donna gets ½ pt.
May 19, 2010
Preference Table for the Smallville Mayoral Election
Number of Votes 130 120 100 150
First Choice A D D C
Second Choice B B B B
Third Choice C C A A
Fourth Choice D A C D
Carmen vs. Donna
Math 132: Foundations of Mathematics
Pairwise Comparison Method• Who wins?
–Antonio: 1 point–Bob: 2½ points–Carmen: 1½ points–Donna: 1 point
• Bob wins! Again!
May 19, 2010
Math 132: Foundations of Mathematics
Who were our winners?
• Plurality: Donna• Borda count: Bob• Plurality-with-elimination: Carmen• Pairwise comparison: Bob• Who should be mayor of Smallville?!?
May 19, 2010
Math 132: Foundations of MathematicsMay 19, 2010
Homework
Page 782: #7Apply all 4 voting methods to determine the kind of play the theater society will perform next semester.
Next Session: Thursday, May 20