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Sagot :
Let's analyze how adding a score of 15 would affect the mean and median scores for both teams.
### Given Values
- Team A: Mean = 13, Median = 12.5
- Team B: Mean = 14, Median = 15
- New Score: 15
### Statements Analysis
#### Statement A: Adding a score of 15 would increase the mean for either team.
- The mean is the average of all scores. Adding a score of 15 will always increase the mean if the new score is higher than the current mean.
- For Team A: [tex]\(13 < 15\)[/tex], hence the mean will increase.
- For Team B: [tex]\(14 < 15\)[/tex], hence the mean will increase.
Therefore, Statement A is correct.
#### Statement B: Adding a score of 15 would increase the median for either team.
- The median is the middle score in an ordered list of scores. Adding a high score like 15 can potentially shift the median upwards.
- For Team A: Current median is 12.5. Adding 15 will either shift the median to 15 or keep it same, either way it will not reduce and since 15>12.5, it will increase the median.
- For Team B: Current median is 15. Adding another 15 will not reduce the median, mean it will remain 15.
Therefore, Statement B is correct.
#### Statement C: Adding a score of 15 to Team A would decrease the difference between the mean scores.
- Initial mean difference between Team A and Team B:
- [tex]\(|13 - 14| = 1\)[/tex]
- New mean for Team A after adding 15:
- New mean = [tex]\((13 + 15) / 2 = 14\)[/tex]
- New mean for Team B after adding 15:
- New mean = 14
So, with both new means being 14, the new mean difference:
- [tex]\(|14 - 14| = 0\)[/tex]
Since [tex]\(0 < 1\)[/tex], the difference decreased.
Therefore, Statement C is correct.
#### Statement D: Adding a score of 15 to Team B would decrease the difference between the mean scores.
- Initial mean difference remains the same as analyzed before, i.e., 1.
- New mean for Team B after adding 15:
- New mean = [tex]\((14 + 15) / 2 = 14.5\)[/tex]
- New mean for Team A remains the same, i.e., 13.
So, the new mean difference:
- [tex]\(|13 - 14.5| = 1.5\)[/tex]
Since [tex]\(1.5 > 1\)[/tex], the difference increased.
Therefore, Statement D is incorrect.
#### Statement E: Adding a score of 15 to Team A would increase the difference between the median scores.
- Initial median difference between Team A and Team B:
- [tex]\(|12.5 - 15| = 2.5\)[/tex]
- New median for Team A after adding 15:
- New median: Will become 15.
- New median for Team B remains the same: 15.
So, the new median difference:
- [tex]\(|15 - 15| = 0\)[/tex]
Since [tex]\(0 < 2.5\)[/tex], the difference decreased instead of increasing.
Therefore, Statement E is incorrect.
### Conclusion
The correct statements are:
- A. Adding a score of 15 would increase the mean for either team.
- B. Adding a score of 15 would increase the median for either team.
- C. Adding a score of 15 to Team A would decrease the difference between the mean scores.
The final list of correct statements is:
- [1, 2, 3]
### Given Values
- Team A: Mean = 13, Median = 12.5
- Team B: Mean = 14, Median = 15
- New Score: 15
### Statements Analysis
#### Statement A: Adding a score of 15 would increase the mean for either team.
- The mean is the average of all scores. Adding a score of 15 will always increase the mean if the new score is higher than the current mean.
- For Team A: [tex]\(13 < 15\)[/tex], hence the mean will increase.
- For Team B: [tex]\(14 < 15\)[/tex], hence the mean will increase.
Therefore, Statement A is correct.
#### Statement B: Adding a score of 15 would increase the median for either team.
- The median is the middle score in an ordered list of scores. Adding a high score like 15 can potentially shift the median upwards.
- For Team A: Current median is 12.5. Adding 15 will either shift the median to 15 or keep it same, either way it will not reduce and since 15>12.5, it will increase the median.
- For Team B: Current median is 15. Adding another 15 will not reduce the median, mean it will remain 15.
Therefore, Statement B is correct.
#### Statement C: Adding a score of 15 to Team A would decrease the difference between the mean scores.
- Initial mean difference between Team A and Team B:
- [tex]\(|13 - 14| = 1\)[/tex]
- New mean for Team A after adding 15:
- New mean = [tex]\((13 + 15) / 2 = 14\)[/tex]
- New mean for Team B after adding 15:
- New mean = 14
So, with both new means being 14, the new mean difference:
- [tex]\(|14 - 14| = 0\)[/tex]
Since [tex]\(0 < 1\)[/tex], the difference decreased.
Therefore, Statement C is correct.
#### Statement D: Adding a score of 15 to Team B would decrease the difference between the mean scores.
- Initial mean difference remains the same as analyzed before, i.e., 1.
- New mean for Team B after adding 15:
- New mean = [tex]\((14 + 15) / 2 = 14.5\)[/tex]
- New mean for Team A remains the same, i.e., 13.
So, the new mean difference:
- [tex]\(|13 - 14.5| = 1.5\)[/tex]
Since [tex]\(1.5 > 1\)[/tex], the difference increased.
Therefore, Statement D is incorrect.
#### Statement E: Adding a score of 15 to Team A would increase the difference between the median scores.
- Initial median difference between Team A and Team B:
- [tex]\(|12.5 - 15| = 2.5\)[/tex]
- New median for Team A after adding 15:
- New median: Will become 15.
- New median for Team B remains the same: 15.
So, the new median difference:
- [tex]\(|15 - 15| = 0\)[/tex]
Since [tex]\(0 < 2.5\)[/tex], the difference decreased instead of increasing.
Therefore, Statement E is incorrect.
### Conclusion
The correct statements are:
- A. Adding a score of 15 would increase the mean for either team.
- B. Adding a score of 15 would increase the median for either team.
- C. Adding a score of 15 to Team A would decrease the difference between the mean scores.
The final list of correct statements is:
- [1, 2, 3]
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