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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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.