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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 5

Question

Math 6010 • Linear Models Fall 2015 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed normal \(N(\mu _1,\sigma ^2)\) random variables. Let \(Y_1,Y_2,\ldots ,Y_m\) be independent identically distributed normal \(N(\mu _2,\sigma ^2)\) random variables. The two samples are independent. The parameters \(\mu _1,\mu _2,\sigma ^2\) are unknown. We wish to test \(H_0:\mu _1=\mu _2\). Show that the two sample t-test and the likelihood ratio test are equivalent.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
normal \(\N(\mu_1,\sigma^2)\) random variables. Let \(Y_1,Y_2,\ldots,Y_m\) be
independent identically distributed normal \(\N(\mu_2,\sigma^2)\) random
variables. The two samples are independent. The parameters \(\mu_1,\mu_2,\sigma^2\)
are unknown. We wish to test \(H_0:\mu_1=\mu_2\). Show that the two sample t-test
and the likelihood ratio test are equivalent.

Topic Tags

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