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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Summer 2020 • Question 7

Question

Math 6010 • Linear Models Summer 2020 • Question 7

Let \(X_1,\ldots ,X_n\) be independent and identically distributed \(N(\mu ,\sigma ^2)\), where \(\mu ,\sigma ^2\) are both unknown. We wish to test \(H_0:\sigma >\sigma _0\) against \(H_a:\sigma \leq \sigma _0\). We reject \(H_0\) if \(S^2<c\), where \[ S^2=\frac {1}{n-1}\sum _{i=1}^n (X_i-\bar X)^2, \qquad \bar X=\frac {1}{n}\sum _{i=1}^n X_i. \] Find \(c\) such that the size of the test is \(\alpha \).

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent and identically distributed
\(\N(\mu,\sigma^2)\), where \(\mu,\sigma^2\) are both unknown. We wish to test
\(H_0:\sigma>\sigma_0\) against \(H_a:\sigma\leq\sigma_0\). We reject \(H_0\) if
\(S^2<c\), where
\[
  S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar X)^2,
  \qquad
  \bar X=\frac{1}{n}\sum_{i=1}^n X_i.
\]
Find \(c\) such that the size of the test is \(\alpha\).

Topic Tags

Covariance and Independence F-Tests and t-Tests Laws of Large Numbers Uniform Integrability and Truncation Methods