Notes:Hypothesis testing
From Maths
- Warning:This page is for my notes on work I haven't done for 5 years, don't rely on it - this is mainly for my own benefit, but if it helps someone else, that's fine by me.
Introduction
A test has two "hypotheses":
- [ilmath]H_0[/ilmath] - the null hypothesis - for now we assign no meaning to it
- [ilmath]H_1[/ilmath] - the alternative hypothesis - a distinct claim from the null
Test outcomes and events
There are 2 outcomes:
- Reject [ilmath]\mathbf{H_1} [/ilmath][Note 1] or "stay with [ilmath]H_0[/ilmath]"
- Accept [ilmath]\mathbf{H_1} [/ilmath] or "discard [ilmath]H_0[/ilmath]"
Despite the recommendation of avoiding terminology like "accept/reject [ilmath]H_0[/ilmath]" it is often used just for symmetry of treatment - remember though that symmetry isn't there!
There are 4 events:
- Correctly reject [ilmath]H_1[/ilmath] / accept [ilmath]H_0[/ilmath]
- Wrongly reject [ilmath]H_1[/ilmath] / accept [ilmath]H_0[/ilmath] - this is sometimes called a type-II error or a type-B error - we denote the probability of such error as [ilmath]\epsilon_b[/ilmath]
- Correctly accept [ilmath]H_1[/ilmath] / reject [ilmath]H_0[/ilmath]
- Wrongly accept [ilmath]H_1[/ilmath] / reject [ilmath]H_0[/ilmath] - this is sometimes called a type-I error or a type-A error - we denote the probability of such an error as [ilmath]\epsilon_a[/ilmath]
Notes
- ↑ Some would say this is the same as accepting [ilmath]H_0[/ilmath], there are many (including me) that voice opposition to this, and would claim you do not accept [ilmath]H_0[/ilmath], but stay with [ilmath]H_0[/ilmath], as you have already accepted [ilmath]H_0[/ilmath] in some way for it to be the null hypothesis.
