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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1999 • Question 7

Question

Math 6010 • Linear Models 1999 • Question 7

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent random samples of independent observations. The distribution of \(X_i\) is normal \(N(\mu _1,\sigma _1^2)\) and the distribution of \(Y_i\) is normal \(N(\mu _2,\sigma _2^2)\). The values of \(\sigma _1\) and \(\sigma _2\) are known. We wish to test \(H_0:\mu _1=\mu _2\) against \(H_A:\mu _1\neq \mu _2\). Show that the likelihood ratio test and the t-test are equivalent under \(H_0\). (20 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) and \(Y_1,Y_2,\ldots,Y_m\) be independent random samples of independent observations. The distribution of \(X_i\) is normal \(N(\mu_1,\sigma_1^2)\) and the distribution of \(Y_i\) is normal \(N(\mu_2,\sigma_2^2)\). The values of \(\sigma_1\) and \(\sigma_2\) are known. We wish to test \(H_0:\mu_1=\mu_2\) against \(H_A:\mu_1\neq\mu_2\). Show that the likelihood ratio test and the t-test are equivalent under \(H_0\). (20 points)

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Topic Tags

F-Tests and t-Tests Likelihood Ratio Tests Maximum Likelihood Estimation Multivariate Normal Distributions