Hypothesis Testing

Hypotheses §

A hypothesis is a statement which makes predictions or has consequences that can be tested. For a hypothesis to be scientifically valid, there must exist an experiment from which a result could be obtained that shows the hypothesis to be false. A hypothesis cannot be proven to be true. However, repeated failure to disprove a hypothesis provides greater evidence for it.

For two random variables, $A$ and $B$, there are several different possible hypotheses:

• $A=B$,
• $A<B$ (lower-tailed test),
• $A\ne B$ (two-tailed test),
• $A>B$ (upper-tailed test).

Defining a Hypothesis Test §

A standard hypothesis test comprises two hypotheses:

• The null hypothesis ($H_0$), which is the assumed state; and
• The alternative hypothesis ($H_1$), which proposes a deviation from the assumed state which can be tested; as well as
• The significance ($\alpha$)-level, which is the required probability for the null hypothesis at which it is rejected in favour of the alternative hypothesis.

The $\alpha$-level can be used to calculate the critical value - the value of the distribution for which the probability is equal to $\alpha$. The critical value is then compared with the test statistic - the observed value being tested - to determine whether or not to reject the null hypothesis.

The confidence interval describes the level of confidence in our result. It is given by $1-\alpha$.Master Method for Recursive Complexities