Students taking an exam are divided into two groups, P and Q such that each group has the same number of students. The performance of each of the students in a test was evaluated out of 200 marks. It was observed that the mean of group P was 105, while that of group Q was 85. The standard deviation of group P was 25, while that of group Q was 5. Assuming that the marks are distributed on a normal distribution, which of the following statements will have the highest probability of being TRUE? A. No student in group Q scored less marks than any student in group P. B. No student in group P scored less marks than any student in group Q. C. Most students of group Q scored marks in a narrower range than students in group P. D. The median of the marks of group P is 100.
GATE 2016 · General Aptitude · Statistics · medium
Answer: Option C: Most students of group Q scored marks in a narrower range than students in group P.
- Evaluate Option A and B (absolute ordering): P's range: 105 - 3*25 = 30 to 105 + 3*25 = 180. Q's range: 85 - 3*5 = 70 to 85 + 3*5 = 100. Overlap exists (70-100 range shared). So A ('no Q student < any P student') and B ('no P student < any Q student') are both false with high probability.
- Evaluate Option D (median of P): Median of P = mean of P = 105 ≠ 100. Option D is false.
- Evaluate Option C (narrower range): sigma_Q = 5 << sigma_P = 25. The 68% interval for Q is [80, 90] (width 10), whereas for P it is [80, 130] (width 50). So most Q students score in a far narrower range than P students. Option C is TRUE.