## What is a Null Hypothesis?

The null hypothesis in the colloquial sense is a **claim that we want to put to the test**. It is usually the default, inherited state of matters, or a state of matters that would be preferred if no further information is available (we do not test). In other words, it should correspond to the position of a critic, skeptical to our proposed alternative or new treatment (the alternative hypothesis). In the **strict statistical sense** of the term it is defined more strictly as a statistical model (see hypothesis).

The null hypothesis is usually denoted by H_{0}.

There are different kinds of null hypotheses, depending on the inference we want to make in the case at hand but in all cases the null and alternative should exhaust the entire parameter space (Θ_{0} ∪ Θ_{1} = Θ, Θ_{0} ∩ Θ_{1} = ∅). Most often in A/B tests we set up a superiority alternative such that the null hypothesis becomes H_{0}: μ ≤ 0 or H_{0}: μ ≤ m where m is some positive discrepancy. However, sometimes the alternative is that of non-inferiority so the null becomes H_{0} ≤ -m where m is the magnitude of the non-inferiority margin.

The null hypothesis should not be confused with the nil hypothesis (no difference hypothesis) although the two can coincide in the case of particular two-sided tests. The null hypothesis can be defined as any point or range which corresponds to a claim that needs to be tested, e.g. μ ≤ 2.

From a practical standpoint the null hypothesis is what gives the p-value a meaningful interpretation since the probability expressed by it is calculated under the assumption that the null is true. Upon observing a p-value below our chosen significance threshold we have sufficient evidence to reject the null hypothesis using the modus tollens logic or the argument from coincidence. We effectively shift the burden of proof to the one who would wand to make a claim corresponding to the null hypothesis.

The null can be **redefined** after a test is completed in order to check how warranted different conclusions are. For example, if you observed a given mean value x and the result is statistically significant for the original H_{0}: μ < ≤ 0 if you were to define H1_{0}: μ ≤ x then you know immediately that you have poor evidence to reject H1_{0} since the probability of observing this value assuming H1_{0} is true is exactly 0.5, thus very likely.

A confidence interval can also be used to test a null hypothesis: every null hypothesis defined over any set of values not covered by the confidence interval can be rejected at significance level equal to 1 minus the confidence level, e.g. if a 95% confidence interval covers the values from 0.01 to +∞ then a null that covers the values from -∞ to 0.005 can be rejected at the 1-0.95 = 0.05 level.