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## ISI MStat 2020 PSB Problem 9

This post discuses the problem 9 of the ISI MStat 2020 PSB Entrance Exam.

A finite population has $N$ units, with $x_{i}$ being the value associated with the $i^{\text {th }}$ unit, $i=1,2, \ldots, N$. Let $\bar{x}{N}$ be the population mean.

A statistician carries out the following experiment.

Step 1: Draw a SRSWOR of size $n({1}$ and denote the sample mean by $\bar{X}{n}$.

Step 2: Draw a SRSWR of size $m$ from $S{1}$. The $x$ -values of the sampled units are denoted by {$Y_{1}, \cdots, Y_{m}$}.

## Hints, Solution, and More

• $\tilde{X}$ follows SRSWOR on population with mean $\mu$.
• $E_{\tilde{X}}\left(\bar{X}{n}\right)=\mu$
• $\tilde{Y} \mid \tilde{X}$ follows SRSWR on $\tilde{X}$ with mean $\bar{X}{n}$
• $E_{\tilde{Y} \mid \tilde{X}}\left(\hat{T}{m}=\bar{Y}{m}\right)=\bar{X}_{n}$
• Use the smoothing property of expectation and variance.

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Learn. Enjoy. Practice. Repeat.

• What will be the case for SRSWR?
• Can you prove the increasing variance idea without doing the variance computation?
• Prove that taking any sampling scheme in the second step, which is unbiased for step 2, will also follow the results, proved above.
• Do practice the SRSWR and SRSWOR variance computation.