## 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.