$H_0$ — Null Hypothesis

- statement that there is no relationship between two variables
- test the likelihood of this statement to be true
- example: $X$ does not impact $Y$
- example: treatment T will not change subjects S in a certain way.

$H_a$ — Alternative Hypothesis

- contradicts the Null hypothesis directly (it’s the opposite)
- The alternative hypothesis states that there is a relationship between two variables
- if you reject $H_0$, then you accept $H_a$.
- The alternative hypothesis is not tested, it’s just the consequence of testing the null hypothesis

# Is it better?

Super cool new treatment will shorten the recovery time $\mu$ from the current standard of 12.3 hours. That’s the alternative hypothesis $H_a$. The null hypothesis $H_0$ is the opposite, or resp. that the treatment is even worse.

$$ H_a : \mu < 12.3 , \text{hours} $$

$$ H_0 : \mu \geq 12.3 , \text{hours} $$

# Does it have any effect at all?

There are some actions or measures or treatment imposed that are beneficial for something else (e.g. cost reductions). Does this action has any effect on a variable we care about? For example does the new treatment change the recovery time $\mu$ compared to the current standard of 12.3 hours? The null hypothesis $H_0$ states that nothing will change.

$$ H_0 : \mu = 12.3 , \text{hours} $$

$$ H_a : \mu \neq 12.3 , \text{hours} $$