# 2. Null Hypothesis and Alternative Hypothesis

## b. distinguish between one-tailed and two-tailed tests of hypotheses;

What is a "null hypothesis" and what is the math notation for it?

- The null hypothesis (designated: `H_0`

), is the statement that is to be tested. The null hypothesis is a statement about the value of a population. The null hypothesis will either be rejected or fail to be rejected.

- The null hypothesis is the "default accepted outcome". It is the thing we already assume to be true and assumes nothing new or interesting is going to happen. E.g., in a court room the defendant is assumed to be innocent.

What is an alternative hypothesis?

- The alternate hypothesis is the statement that is accepted if the sample data provides sufficient evidence that the null hypothesis is false. It is designated as `H_1`

and is accepted if the sample data provides sufficient statistical evidence that `H_0`

is false.

- The alternative hypothesis is often what you want to prove, but you have to prove that the null hypothesis is false in order to do it. E.g. in a court room the null hypothesis is that the defendant is innocent, and the prosecution has an alternative hypothesis that the defendant is guilty and has to prove that the null hypothesis is false.

Why is the null hypothesis called "null"?

"Null" is shorthand for "nullify". It is a hypothesis that you can nullify and invalidate.

If you accept the null hypothesis does that mean it is true?

No. It simply means there is not enough evidence to reject it.

What is a two-tailed hypothesis?

- If the null hypothesis has a specific value and the alternative hypothesis can be either greater or less than the null, then it is two-tailed. E.g.

- $$H_0: \mu = \mu_0$$ versus $$H_1: \mu \neq \mu_0$$

What is a one-tailed hypothesis?

- If the null hypothesis tests whether there is evidence that the actual parameter is significantly higher (or lower) than the hypothesized value. The alternative hypothesis can only be greater or less than the null hypothesis. E.g.

- $$H_0: \mu \leq \mu_0$$ versus $$H_1: > \mu_0$$

##### Source:

- CFA