QB_PH_123: Bose-Einstein and Fermi-Dirac statistics

For a random sample of size $n = 37$ drawn from a normal distribution $\mathcal{N}(\mu, \sigma^2)$ with known standard deviation $\sigma = 5$, what is the standard error of the sample mean $\bar{X}$?

Correct Answer

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Fundamentals of Bose-Einstein and Fermi-Dirac statistics** under *Bose-Einstein and Fermi-Dirac statistics* in GATE PH. 2. **Stage 2: Mathematical Derivation:** * Sampling distribution of the mean: $\bar{X} \sim \mathcal{N}\left(\mu, \frac{\sigma^2}{n}\right)$. Standard error is the square root of the variance: $\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{37}} = 0.82$. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Dividing by $n$ instead of $\sqrt{n}$. *Reference: R.K. Pathria, Paul D. Beale, Statistical Mechanics (Academic Press).*