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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 5

Question

Math 6010 • Linear Models Fall 2017 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be iid \(N(\mu _1,\sigma _1^2)\) and \(Y_1,Y_2,\ldots ,Y_m\) be iid \(N(\mu _2,\sigma _2^2)\). The \(X\) and the \(Y\) are also independent of each other. Assume that \(\sigma _1^2,\sigma _2^2\) are known while \(\mu _1,\mu _2\) are unknown. We wish to test \(H_0:\mu _1=\mu _2\) against \(H_A:\mu _1\ne \mu _2\). Find a test using the likelihood method. Compute the distribution of the likelihood ratio test.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid \(\N(\mu_1,\sigma_1^2)\) and
\(Y_1,Y_2,\ldots,Y_m\) be iid \(\N(\mu_2,\sigma_2^2)\). The \(X\) and the \(Y\) are also
independent of each other. Assume that \(\sigma_1^2,\sigma_2^2\) are known while
\(\mu_1,\mu_2\) are unknown. We wish to test \(H_0:\mu_1=\mu_2\) against
\(H_A:\mu_1\ne\mu_2\). Find a test using the likelihood method. Compute the
distribution of the likelihood ratio test.

Topic Tags

Covariance and Independence F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation