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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1999 • Question 3

Question

Math 6010 • Linear Models 1999 • Question 3

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed normal \(N(0,\sigma ^2)\) random variables. We wish to test \[ H_0:\sigma \leq \sigma _0 \] against \[ H_A:\sigma >\sigma _0. \]

  1. Find the uniformly most powerful test of size \(\alpha \). (10 points)
  2. Compute the rejection region as a function of \(\alpha \). (10 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed normal \(N(0,\sigma^2)\) random variables. We wish to test
\[
H_0:\sigma\leq\sigma_0
\]
against
\[
H_A:\sigma>\sigma_0.
\]
\begin{enumerate}
\item Find the uniformly most powerful test of size \(\alpha\). (10 points)
\item Compute the rejection region as a function of \(\alpha\). (10 points)
\end{enumerate}

\newpage

Topic Tags

Covariance and Independence F-Tests and t-Tests Measure-Theoretic Probability Uniform Integrability and Truncation Methods