Information

Question

The mean and median scores of the two teams playing a game are shown in the table.



Team Mean Median
A 13 12.5
B 14 15

Question
A new team member expects to be able to score 15 points. The teams are studying how including her score would affect the teams.



Choose ,begin emphasis,all,end emphasis, correct statements about how adding a score of 15 would affect the mean and median scores.

Answer options with 5 options
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.

D.
Adding a score of 15 to Team B would decrease the difference between the mean scores.

E.
Adding a score of 15 to Team A would increase the difference between the median scores.

Answer 1

Let's analyze the statements one by one based on the current mean and median scores of both teams (Team A and Team B) and the impact of adding a score of 15.

Current Scores

• Team A: Mean = 13, Median = 12.5
• Team B: Mean = 14, Median = 15

Impacts of Adding a Score of 15

1. Team A:

   • Assuming Team A has a total of $n$ players, the total score is $13n$.
   • Adding a score of 15: New total score = $13n+15$, new mean = $(13n+15)/(n+1)$.
   • Since 15 is greater than the current mean of 13, the new mean will increase.
   • The new median will depend on the distribution of the current scores. Since the current median is 12.5, adding a 15 likely will increase the median if there are currently less than half the scores below 15.

2. Team B:

   • Assume Team B has $m$ players, with a total score of $14m$.
   • Adding a score of 15: New total score = $14m+15$, new mean = $(14m+15)/(m+1)$.
   • Since 15 is equal to the current mean of 14, the new mean will increase slightly.
   • The median remains at 15 since the new score does not change the existing order unless there are fewer than half of the players scoring above 15.

Evaluating the Statements:

A. Adding a score of 15 would increase the mean for either team.

• True. Adding 15 increases the mean for both teams.

B. Adding a score of 15 would increase the median for either team.

• Uncertain/True for Team A: Likely true for Team A, but depends on the distribution. Team B's median remains at 15, so not true for them.

C. Adding a score of 15 to Team A would decrease the difference between the mean scores.

• True. The mean for Team A will increase, and since the difference initially is $14-13=1$, it will decrease as Team A’s score increases.

D. Adding a score of 15 to Team B would decrease the difference between the mean scores.

• True. Since Team B's mean will slightly increase, the difference with Team A's mean will decrease.

E. Adding a score of 15 to Team A would increase the difference between the median scores.

• Uncertain/False. The median for Team A is likely to increase, potentially decreasing the difference since Team B's median is already at 15.

Correct Statements:

Thus, the correct statements regarding the impact of including a score of 15 are:

• A (True)
• C (True)
• D (True)

So the final answers would be options A, C, and D.