GATE 2021 CH – Question 45
The probability distribution function of a random variable X is shown in the figure. From this distribution, random samples with sample size $n=68$ are taken. If $\bar X$ is the sample mean, the standard deviation of the probability distribution of $\bar X$, i.e. $\sigma_{\bar X}$ is _____ (round off to 3 decimal places).
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Correct answer: 0.060 to 0.080
Explanation
The rectangular density represents $X\sim U(1,3)$. Its variance is $(b-a)^2/12=(3-1)^2/12=1/3$.
For the mean of 68 independent observations, variances divide by the sample size:
$$\operatorname{Var}(\bar X)=\frac{1/3}{68}=\frac1{204},\qquad \sigma_{\bar X}=\frac1{\sqrt{204}}=0.0700140.$$
Thus the requested standard deviation is **0.070**. It is the standard error of the mean, not the original distribution’s standard deviation.