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                                    "content": "2026 MCM Problem C: Data With The Stars"
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                                    "content": "Dancing with the Stars (DWTS) is the American version of an international television franchise based on the British show \"Strictly Come Dancing\" (\"Come Dancing\" originally). Versions of the show have appeared in Albania, Argentina, Australia, China, France, India, and many other countries. The U.S. version, the focus of this problem, has completed 34 seasons."
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                                    "content": "Celebrities are partnered with professional dancers and then perform dances each week. A panel of expert judges scores each couple's dance, and fans vote (by phone or online) for their favorite couple that week. Fans can vote once or multiple times up to a limit announced each week. Further, fans vote for the star they wish to keep, but cannot vote to eliminate a star. The judge and fan votes are combined in order to determine which couple to eliminate (the lowest combined score) that week. Three (in some seasons more) couples reach the finals and in the week of the finals the combined scores from fans and judges are used to rank them from "
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                                    "content": "There are many possible methods of combining fan votes and judge scores. In the first two seasons of the U.S. show, the combination was based on ranks. Season 2 concerns (due to celebrity contestant Jerry Rice who was a finalist despite very low judge scores) led to a modification to use percentages instead of ranks. Examples of these two approaches are provided in the Appendix."
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                                    "content": "In season 27, another \"controversy\" occurred when celebrity contestant Bobby Bones won despite consistently low judges scores. In response, starting in season 28 a slight modification to the elimination process was made. The bottom two contestants were identified using the combined judge scores and fan votes, and then during the live show the judges voted to select which of these two to eliminate. Around this same season, the producers also returned to using the method of ranks to combine judges scores with fan votes as in seasons one and two. The exact season this change occurred is not known, but it is reasonable to assume it was season 28."
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                                            "content": "- How much certainty is there in the fan vote totals you produced, and is that certainty always the same for each contestant/week? Provide measures of your certainty for the estimates."
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                                            "content": "○ Compare and contrast the results produced by the two approaches used by the show to combine judge and fan votes (i.e. rank and percentage) across seasons (i.e. apply both approaches to each season). If differences in outcomes exist, does one method seem to favor fan votes more than the other?"
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                                            "content": "○ Examine the two voting methods applied to specific celebrities where there was “controversy”, meaning differences between judges and fans. Would the choice of method to combine judge scores and fan votes have led to the same result for each of these contestants? How would including the additional approach of having judges choose which of the bottom two couples to eliminate each week impact the results? Some examples you might consider (there may also be others you identified):"
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                                            "content": "- season 2 - Jerry Rice, runner up despite the lowest judges scores in 5 weeks."
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                                    "content": "- Based on your analysis, which of the two methods would you recommend using for future seasons and why? Would you suggest including the additional approach of judges choosing from the bottom two couples?"
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                                            "content": "- Use the data including your fan vote estimates to develop a model that analyzes the impact of various pro dancers as well as characteristics for the celebrities available in the data (age, industry, etc). How much do such things impact how well a celebrity will do in the competition? Do they impact judges scores and fan votes in the same way?"
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                                            "content": "- Propose another system using fan votes and judge scores each week that you believe is more \"fair\" (or \"better\" in some other way such as making the show more exciting for the fans). Provide support for why your approach should be adopted by the show producers."
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                                            "content": "- Your complete solution."
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                                            "content": "7. A 0 score is recorded for celebrities who are eliminated. For example, in Season 1 the first celebrity eliminated was Trista Sutter at the end of the Week 2 show. She thus has scores of 0 for the rest of the season (week 3 through week 6)."
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                                    "content": "Appendix: Examples of Voting Schemes"
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                                    "content": "2. COMBINED BY PERCENT (used for season 3 through "
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                                    "content": "Starting in season 3 scores were combined using percents instead of ranks. An example is shown using week 9 of season 5. In that week, Jennie Garth was eliminated. Again, we artificially created fan votes that produce total percents to correctly lead to that result. The judges' percent is computed by dividing the total judge score for the contestant by the sum of total judge scores for all 4 contestants. Based on the judges' percent, Jennie was "
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                                            "content": "Table 3: Example of Combining Judge and Fan Votes by Percent (Season 5, Week 9)"
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                                            "html": "<table><tr><td>Contestant</td><td>Total Judges Score</td><td>Judges Score Percent</td><td>Fan Vote*</td><td>Fan Percent*</td><td>Sum of Percent</td></tr><tr><td>Jennie Garth</td><td>29</td><td>29/117 = 24.8%</td><td>1.1 million</td><td>1.1/10 = 11%</td><td>35.8</td></tr><tr><td>Marie Osmond</td><td>28</td><td>28/117 = 23.9%</td><td>3.7 million</td><td>3.7/10 = 37%</td><td>60.9</td></tr><tr><td>Mel B</td><td>30</td><td>30/117 = 25.6%</td><td>3.2 million</td><td>3.2/10 = 32%</td><td>57.8</td></tr><tr><td>Helio Castroneves</td><td>30</td><td>30/117 = 25.6%</td><td>2 million</td><td>2/10 = 20%</td><td>45.6</td></tr><tr><td>Total</td><td>117</td><td></td><td>10 million</td><td></td><td></td></tr></table>",
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