Скачать или смотреть UMP Test for Variance in Normal Distribution | UPSC ISS 2024 Paper-2 | Problem-17 | RitwikMath

This video discusses hypothesis testing about the variance \(\sigma^2\) of a normal distribution \(N(0, \sigma^2)\) using the uniformly most powerful (UMP) test. Given a sample \(X_1, X_2, \ldots, X_n\), the critical region is expressed as
\[
\sum_{i=1}^n X_i^2 \geq 2 k_\alpha^2,
\]
where \(k_{\alpha}^2\) is the critical value derived from the chi-square distribution with \(n\) degrees of freedom such that
\[
P(\chi^2_n \geq k_\alpha^2) = \alpha.
\]

The video explains that the test is one-sided for higher variance alternatives, rejecting when the test statistic exceeds the threshold, confirming option A as the correct choice. Lower-tail tests and other options do not fit the critical region or test design correctly. This analysis is essential for understanding variance testing in parametric inference and UPSC exam preparation.

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