A group of 8 students played a certain game. Every player received a score equal to an integer from 1 to 10, inclusive.
For the 8 players, the mean score was 6. If more than half of the players received a score greater
than 7, which of the following is true about the mean score of the remaining players?
We are given a mean score of 6. If more than half the players received a score greater than 7, our intuition
should tell us that the other scores must be rather low to 'cancel out' the higher scores.
Only one option includes low scores for the remaining players.
We can use the given information to create a sample set of scores. Since more than half the players received a score greater than 7, one option is:
{8,8,8,8,8, a,b,c}
Where \(a,b, \text{ and } c\) represent the other three students.
The mean for all of the students is 6, which means that the total score is
$$ 6   (\text{average}) \cdot 8 \text{ students} $$
$$ =48 \text{ points} $$
We can sum up the scores of our sample set:
$$ 8+8+8+8+8+a+b+c=48 $$
$$ a+b+c=8 $$
The mean of these three numbers is represented by:
$$ \frac{a+b+c}{3}=\frac{8}{3} $$
The mean for our sample set for the other students is \(\dfrac{8}{3} \), which is \( \boxed{\text{less than 6}}\).