Hypothesis

a hypothesis is a scientific question posed in such a way as it can be tested.

typically these are a statement about a particular value, or range of values for some variable or parameter.

The entire parameter space is referred to as the event space and is denoted $$\Omega$$

So a hypothesis can be formulated that

$$\theta \in \omega$$ where $$\omega$$ is a subset of $$\Omega$$

ie that $$\theta$$ is in some range, or list of values $$\omega$$ from the total possible event space $$\Omega$$

Parameter Space
the parameter space is dictated by how the question is defined, for example for a ratio, like gender-at-birth ratio can take any non-negative value.

$$\Omega = [0,\infty)$$

So it would be possible to construct a hypothesis that the number of make and female are identical and hence the ratio is 1, $$H:\theta=1$$

For the mean of the normal distribution the parameter could conceivably take any value on the real number line like so;

$$\Omega = (-\infty,\infty)$$

Simple hypothesis
that $$\theta$$ takes a single value is referred to as a simple hypothesis

hypothesis where there is more than one value of $$\theta$$ is called a composite hypothesis

The null hypothesis
this is sometime referred to as a privileged hypothesis, $$H_0 $$ and has a complement, the alternative hypothesis $$H_1 $$

the alternative hypothesis
$$H_1$$ can be written $$H_1: \theta \notin \omega $$

null and alternative hypothesis as parameter space
The null hypothesis is denoted with the zero subscript like so $$H_{0}:\theta = 0$$

$$H_0:\theta \in \omega$$ then $$H_1:\theta \in \Omega - \omega$$

(is is also accurate to put) $$\notin \omega$$