Definition:Significance Level

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Definition

Let $\delta$ be a hypothesis test.

Let $H_0$ and $H_1$ be the null hypothesis and alternative hypothesis of $\delta$ respectively.

Let $T$ be the test statistic which is being used to determine whether $H_0$ or $H_1$ holds.

Let $C$ be the critical region of $\delta$.

Let $\alpha$ be the probability that $T$ takes a value in $C$ when in fact $H_0$ holds.


$\alpha$ is known as the significance level of $\delta$.


Examples

Significance levels $\alpha$ are traditionally taken to be one of the following:

\(\ds \alpha\) \(=\) \(\ds 0 \cdotp 05\) that is, $5 \%$
\(\ds \alpha\) \(=\) \(\ds 0 \cdotp 01\) that is, $1 \%$
\(\ds \alpha\) \(=\) \(\ds 0 \cdotp 001\) that is, $0 \cdotp 1 \%$


Motivation

Until relatively recently, these significance levels have been the only ones which were used in practice.

This is because, prior to the advent of high speed electronic computation devices, the process to calculate the $p$-value was time-consuming and generally unfeasible.

Hence critical values corresponding to the above significance levels have been tabulated for many of the commonly used test statistics, such as for the $t$-test, the $F$-test and the $\chi$-squared test.


However, modern statistical software makes it easy to calculate an exact value of $p$.

Hence it is possible to achieve a greater level of information than just to state that $p$ lies between two of the above bounds.


However, these conventional significance levels can still be considered useful criteria for establishing significance.


Also see

  • Results about significance levels can be found here.


Sources

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