Null hypothesis

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In Statistics, the null hypothesis, represented by $H_0$ , is a hypothesis that is presented on certain statistical facts and whose falsity is tried to be proved through an adequate test of hypotheses.

A null hypothesis generally states that there is no relationship between two measured phenomena. For example, in physics: "increasing the potential difference does not affect the current in a conductor". In pharmacological research: "a potential medical treatment has no effect". In market research: "a 5% increase in the price of a particular product will not adversely affect sales of that product".

It is a hypothesis that is intended to be confronted with the facts. It is often a statement about a parameter that is a statistical property of a population. When it is not possible or feasible to observe the entire population, the test is based on the observation of a random sample of the population. Such a parameter is often the mean or standard deviation.

Often such a hypothesis consists of stating that the parameters or mathematical characteristics of two or more populations are identical.

Normally, the null hypothesis is formulated in the form of an equality, that is, it is a simple hypothesis.

Example

If we want to compare the test results of two random samples of men and women, the null hypothesis could be "that the mean of the test result of the male population from which the first sample was taken is the same as that of the female population sample, of the second sample": ${displaystyle H_{0}:mu _{1}=mu _{2}}$

on what:H0 ₌ the null hypothesisμ ₁ = the population mean 1, andμ ₂ = the population mean 2.

Alternatively, the null hypothesis can posit that the two samples are drawn from the same population: ${displaystyle H_{0}:mu _{1}-mu _{2}=0}$

Possible conclusions about the null hypothesis

As mentioned, a hypothesis test tries to prove the falsity of the null hypothesis. Thus, only two decisions can be made:

reject the null hypothesis;
not reject the null hypothesis.

It should be noted that not rejecting the null hypothesis just means that it was not possible, through the available data, to demonstrate its falsity, which is completely different from proving its veracity.

Making an analogy with court proceedings, the null hypothesis is that the defendant is innocent. During the trial, an attempt is made to prove the falsity of this hypothesis, that is, that the defendant is guilty. But in the event that guilt cannot be proved, this does not mean that the defendant is innocent; it just means that not enough evidence has been found. It is completely different, therefore, not to reject a hypothesis or to accept a hypothesis.

The fact that the null hypothesis cannot be accepted, but only not rejected, has to do with the mistakes that can be made when rejecting or not rejecting the hypothesis.

Null hypothesis and alternative hypothesis

In most cases, at the same time as the null hypothesis is formulated, another hypothesis is formulated, contradictory to the first one, called the alternative hypothesis, denoted by $H_1$ . The alternative hypothesis, however, does not have to be the negation of the null hypothesis. For example, suppose we formulate the following null hypothesis about the value of the population mean:

H ₀: μ = θ

We can formulate the following alternative hypotheses:

H ₁: μ ≠ θ
H ₁: μ < θ
H ₁: μ > θ

Each null hypothesis/alternative hypothesis pair leads to a different hypothesis test. It should be noted that assuming a different alternative hypothesis may lead to a different decision regarding the null hypothesis.

The choice of the null hypothesis/alternative hypothesis depends on the context of the problem, the parameter to be tested and the conclusions to be reached, taking into account that the null hypothesis is always formulated in the form of equality.

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