There are \(9\) people in the Pin Pals bowling club. After a game one night, the scores for each player were \(157\), \(153\), \(159\), \(161\), \(153\), \(151\), \(156\), \(157\), and \(154\). During the game, the players were split between two lanes: Lane 1 and Lane 2. The following information is known about the scores of the players in each of the lanes.
No player in Lane 1 scored higher than any player in Lane 2. However, one player in Lane 1 had the same score as a player in Lane 2.
Two of the players in Lane 1 had the same score.
Determine the difference between the mean score of the players in Lane 1 and the mean score of the players in Lane 2.
This problem was inspired by a past Beaver Computing Challenge (BCC) problem.
We start by placing the scores in ascending order:
\(151\), \(153\), \(153\), \(154\), \(156,\), \(157\), \(157\), \(159\), \(161\)
Since no player in Lane 1 scored higher than any player in Lane 2, the scores for the players in Lane 1 will be the numbers on the left of this list, and the scores for the players in Lane 2 will be the numbers on the right. We need to determine where in this list the scores change from Lane 1 to Lane 2.
We are told that one player in Lane 1 had the same score as a player in Lane 2. There are two possibilities for this. Either these players both had scores of \(153\) or they both had scores of \(157\).
Similarly, we are told two of the players in Lane 1 had the same score. Then either these players both had scores of \(153\) or they both had scores of \(157\). We will consider these two cases.
If two players in Lane 1 had scores of \(157\), then since \(153 < 157\), the two players with scores of \(153\) must also both have been in Lane 1. However this contradicts the fact that one player in Lane 1 had the same score as a player in Lane 2. So two players in Lane 1 could not have had scores of \(157\).
If two players in Lane 1 had scores of \(153\), then it’s possible one of the players with a score of \(157\) was in Lane 1 and the other was in Lane 2. In this case, the players in Lane 1 scored \(151\), \(153\), \(153\), \(154\), \(156\), and \(157\). Then, the players in Lane 2 scored \(157\), \(159\), and \(161\). Since there are no other cases, this must be the solution.
The mean score for a player in Lane 1 is \(\dfrac{151+153+153+154+156+157}{6}=\dfrac{924}{6}=154\).
The mean score for a player in Lane 2 is \(\dfrac{157+159+161}{3}=\dfrac{477}{3}=159\).
Therefore, the difference between the mean score of the players in Lane 1 and the mean score of the players in Lane 2 is \(159-154=5\).