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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 {\(Y_{1}, \cdots, Y_{m} \)}.

- \(\tilde{X}\) 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.

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