## 4.3 Putting it all together

Having discussed the $$t$$-test assumptions and checking for normality, we will now summarise the discussion with some simple guidelines for the purposes of this subject.

What do you need to know?

1. What the $$t$$-test assumptions are
2. How to check for Normality using the three ways presented in this chapter
3. How to make a decision as to whether or not the normality assumption has been violated.

For the purposes of this subject, use the following rules to guide the decision as to whether or not the normality assumption has been violated:

Normality assumption decision:

• If the underlying distribution is normal, then the distribution of the sample mean will also be normal. This means the normality assumption has not been violated and the $$t$$-test can be used. This is regardless of sample size.
• If the underlying distribution is not normal but $$n \geq 30$$, then the distribution of the sample mean will be at least approximately normal. This means the normality assumption has not been violated and the $$t$$-test can be used.
• If the underlying distribution is not normal and $$n < 30$$, then we cannot assume that the distribution of the sample mean is normal. This means the normality assumption has been violated and we should not use the $$t$$-test.

What happens when the normality assumption has been violated?

If the normality assumption has been violated, we can use what is called a 'non-parametric' test. That is, a test that does not make any assumptions about the underlying distribution of the data. However, these types of tests are beyond the scope of this subject.