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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 2

Question

Math 6010 • Linear Models Fall 2007 • Question 2

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be two independent random samples. We assume that \(X_1,X_2,\ldots ,X_n\) are independent identically distributed normal \((\mu _1,\sigma ^2)\) random variables and \(Y_1,Y_2,\ldots ,Y_m\) are independent identically distributed normal \((\mu _2,\sigma ^2)\) random variables (\(\sigma ^2\) is unknown). We want to test \(H_0:\mu _1=\mu _2\) against the alternative \(H_A:\mu _1\ne \mu _2\).

Show that the likelihood ratio test and the two sample t-test are equivalent. (7 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) and \(Y_1,Y_2,\ldots,Y_m\) be two
independent random samples. We assume that \(X_1,X_2,\ldots,X_n\) are
independent identically distributed normal \((\mu_1,\sigma^2)\) random variables
and \(Y_1,Y_2,\ldots,Y_m\) are independent identically distributed normal
\((\mu_2,\sigma^2)\) random variables (\(\sigma^2\) is unknown). We want to test
\(H_0:\mu_1=\mu_2\) against the alternative \(H_A:\mu_1\ne\mu_2\).

Show that the likelihood ratio test and the two sample t-test are equivalent.
(7 points)

Topic Tags

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