6.1 Statistical Hypothesis: General Concepts

Definition

A statistical hypothesis is an assertion or conjecture concerning one or more populations.

6.1.1 The Null and Alternative Hypothesis

In hypothesis testing, we have the null hypothesis $H_0$ which is some hypothesis we wish to find evidence against. Rejection of $H_0$ leads to the acceptance of an alternative hypothesis $H_1$.

$H_0$ states that a population parameter is equal to a value $\theta_0$.

$H_1$ states that a population parameter is not equal to a value $\theta_0$.

$$ H_0:\theta=\theta_0\\ H_1:\theta\neq\theta_0 $$

In all hypothesis tests, we will arrive at one of the two following conclusions:

1. Reject $H_0$ in favor of $H_1$ because of sufficient evidence in the data, or
2. Fail to reject $H_0$ because of insufficient evidence

Note

The conclusions do not say “accept $H_0$”. Failing to find evidence against $H_0$ means only that the data are consistent with $H_0$, not that we have clear evidence that $H_0$ is true. So we can only reject or fail to reject $H_0$.

Example 1

6.1.2 One- and Two-Tailed Test

Any statistical hypothesis where the alternative is one sided (”>”), e.g.