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
standard deviation of group P was 25, while that of group Q was 5. Assuming that the marks were
distributed on a normal distribution, which of the following statements will have the highest
probability of being TRUE?
Options:
(A) Antisense oligonucleotide
(B) Ribozyme
(C) Aptamer
(D) Avidin
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2016 Q.06)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Statistics & Probability (Properties of Normal Distribution & Standard Deviation)
**2. Detailed Step-by-Step Analysis**:
- The performance marks of two student groups, $P$ and $Q$, of identical cohort size follow normal distributions:
- Group $P$: Standard deviation $\sigma_P = 25$.
- Group $Q$: Standard deviation $\sigma_Q = 5$.
- In a normal distribution, standard deviation ($\sigma$) measures the dispersion/spread of data points around the mean ($\mu$):
- Approximately $68.2\%$ of scores fall within $[\mu - \sigma, \mu + \sigma]$.
- Approximately $95.4\%$ of scores fall within $[\mu - 2\sigma, \mu + 2\sigma]$.
- For Group $Q$, with a much smaller standard deviation ($\sigma_Q = 5$), the distribution curve is tall and narrow (leptokurtic relative to P). Scores are tightly clustered around the mean within a narrow band of $\pm 10$ to $\pm 15$ marks.
- For Group $P$, with $\sigma_P = 25$, scores are dispersed widely over a broad range of $\pm 50$ to $\pm 75$ marks.
- Therefore, there is a very high probability that most students in group $Q$ scored marks within a much narrower range than students in group $P$.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Group means are not given; if $\mu_P > \mu_Q$, many students in $P$ score higher than $Q$.
- **Option (B) [INCORRECT]**: Cannot be asserted without knowing the means.
- **Option (C) [CORRECT]**: Since $\sigma_Q = 5 \ll \sigma_P = 25$, the marks in group $Q$ are concentrated in a much narrower range.
- **Option (D) [INCORRECT]**: The mean/median is not specified to be 100.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **C**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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