Hypothesis Testing

The main problems in statistical inference can be broadly classified into two areas:

• The area of estimation of population parameters and setting up confidence intervals for them, ie., the area of point and interval estimation and
• Tests of statistical hypothesis.

For example: If ${\mathit{X}}_1,{\mathit{X}}_2,...,{\mathit{X}}_{\mathit{n}}$ is a random sample of size $\mathit{n}$ from a normal population with mean $\mathit{\mu }$ and variance ${\mathit{\sigma }}^2$ then the hypothesis ${\mathit{H}}_0:\mathit{\mu }={\mathit{\mu }}_0$, ${\mathit{\sigma }}^2={\mathit{\sigma }}_0^2$ is a simple hypothesis, whereas each of the following is the composite hypothesis:

1. $\mathit{\mu }={\mathit{\mu }}_0$
2. ${\mathit{\sigma }}^2={\mathit{\sigma }}_0^2$
3. $\mathit{\mu }<{\mathit{\mu }}_0,{\mathit{\sigma }}^2={\mathit{\sigma }}_0^2$
4. $\mathit{\mu }>{\mathit{\mu }}_0,{\mathit{\sigma }}^2={\mathit{\sigma }}_0^2$
5. $\mathit{\mu }={\mathit{\mu }}_0,{\mathit{\sigma }}^2<{\mathit{\sigma }}_0^2$
6. $\mathit{\mu }={\mathit{\mu }}_0,{\mathit{\sigma }}^2>{\mathit{\sigma }}_0^2$
7. $\mathit{\mu }<{\mathit{\mu }}_0,{\mathit{\sigma }}^2>{\mathit{\sigma }}_0^2$

A hypothesis that doesn't specify completely $\text{'}\mathit{r}\text{'}$ parameter of a population is termed as a composite hypothesis with $\mathit{r}$ degree of freedom.

Test of a Statistical Hypothesis

A test of statistical hypothesis is a two-action decision problem after the experimental sample values have been obtained the two actions being the acceptance or rejection of the hypothesis under consideration.

Null Hypothesis$\left({\mathit{H}}_0\right)$

This can be thought of as the implied hypothesis. “Null” means “nothing.”  This hypothesis states that there is no difference between groups or no relationship between variables. The null hypothesis is a presumption of status quo or no change.

Alternative Hypothesis$\left({\mathit{H}}_{\mathit{A}}\right)$ 

This is also known as the claim. This hypothesis should state what you expect the data to show, based on your research on the topic. This is your answer to your research question.