Significance Level
The significance level is a threshold used in statistics to determine how unlikely an observed result must be before rejecting the null hypothesis. It is represented by \( \alpha \).
It indicates the probability of committing a Type I error, which is the chance of rejecting the null hypothesis when it is, in fact, true.
In simpler terms, the significance level sets the decision rule for accepting or rejecting a hypothesis based on the probability of error one is willing to accept.
What is a Type I error? A Type I error happens in inferential statistics when I reject the null hypothesis (H0) that is actually true. In other words, it’s a false positive. I conclude that there is an effect or a relationship between two or more variables when, in reality, there isn’t one.
What does this mean in practice?
When I conduct a statistical test (such as a chi-square test), I’m evaluating if the observations align with the null hypothesis.
For instance, a null hypothesis (H0) might state that two statistical variables \( X \) and \( Y \) are independent, meaning that changes in one don’t influence the other.
In this scenario, the significance level defines the threshold beyond which I determine that the observed outcome is not due to random chance but rather indicates a genuine difference or a relationship between the two variables.
How do you choose a significance level?
A common significance level is \( \alpha = 0.05 \), meaning there’s a 5% probability of committing a Type I error.
In other words, I accept the risk that the result could be due to chance 1 time out of 20.
Other typical values are \( \alpha = 0.01 \) (1%) and \( \alpha = 0.10 \) (10%), depending on the context and the level of precision needed.
How is the significance level applied?
In statistical independence tests like the chi-square \( \chi^2 \) test, the significance level, together with the degrees of freedom, helps calculate the critical value to compare with the chi-square statistic.
- If the test statistic exceeds the critical value, I reject the null hypothesis. This indicates that the result is statistically significant, suggesting it’s unlikely the observations are due to chance.
- If the test statistic does not exceed the critical value, there isn’t enough evidence to reject the null hypothesis, so I conclude that the result is not statistically significant. The variables are likely independent.
Example: I apply the chi-square test to check for an association between two variables. I choose a significance level of \( \alpha = 0.05 \), and the contingency table has 2 degrees of freedom. Looking up the chi-square distribution table, the critical value is 5.99. If the \( \chi^2 \) value I calculate exceeds 5.99, I reject the null hypothesis and conclude that there is a relationship between the variables. If it’s less than or equal to 5.99, I accept the null hypothesis, indicating there is not enough evidence to establish a relationship between the variables.
And so forth.
