1.4 Other types of alternative hypotheses: Independent samples $t$-test

In our discussion of the independent samples $t$-test, we have only considered the two-sided hypothesis test:

$$H_0:\mu_1 = \mu_2\;\;\text{versus}\;\;H_1:\mu_1 \neq \mu_2,$$

or equivalently,

$$H_0:\mu_1 - \mu_2 = 0\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 \neq 0.$$

One-sided tests are also possible. For example,

$$H_0:\mu_1 - \mu_2 = 0\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 < 0.$$

or

$$H_0:\mu_1 - \mu_2 = 0\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 > 0.$$

In addition, we can test for differences between groups that are greater, less than, or not equal to a specified value other than 0. For example:

$$H_0:\mu_1 - \mu_2 = 5\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 \neq 5,$$

or

$$H_0:\mu_1 - \mu_2 = 5\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 < 5,$$

or

$$H_0:\mu_1 - \mu_2 = 5\;\;\text{versus}\;\;H_1:\mu_1 - \mu_2 > 5.$$

Although we may have a chance to see how these additional types of hypotheses work in the computer lab, you will not be required to use them for the purposes of this subject.

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND