Q12
Eight employees of an organization have been rated on a scale of 1 to 50 for their performance. All ratings are integers. The overall average rating of the eight employees is 30. While the five employees with the highest ratings average 38, the five employees with the lowest ratings average 25.
Which of the following, about the ratings obtained by the eight employees, is DEFINITELY FALSE?
Answer: E
Asked: which statement about the 8 ratings is definitely false.
Sort the ratings: \(r_1 \le r_2 \le \dots \le r_8\).
- Total: \(8 \times 30 = 240\)
- Top five: \(r_4 + \dots + r_8 = 5 \times 38 = 190\)
- Bottom five: \(r_1 + \dots + r_5 = 5 \times 25 = 125\)
- \(r_4\) and \(r_5\) are counted in both sums, so \(r_4 + r_5 = 190 + 125 - 240 = 75\).
- So \(r_1 + r_2 + r_3 = 50\) and \(r_6 + r_7 + r_8 = 115\).
- Since \(r_4 \le r_5\) and they add to 75: \(r_5 \ge 38\).
Statement 5 (highest = 40): then \(r_6 + r_7 = 75\). But \(r_6, r_7 \ge r_5 \ge 38\) means \(r_6 + r_7 \ge 76\). This is impossible, so statement 5 is definitely false.
The others can hold:
- Statement 1: 37, 38, 38, 38, 39 at the top works (\(r_8 = 39\)).
- Statement 2: e.g. 1, 12, 37, 37, 38, ...
- Statement 3: e.g. 6, 7, 37, 37, 38, ...
- Statement 4: the median is always \(\frac{r_4 + r_5}{2} = 37.5\).
Answer: The highest rating obtained is 40.