What happens to the probability of making a Type II error, β, as the level of significance, α, decreases? Why?

Verified step by step guidance

1. Recall that the level of significance, \( \alpha \), is the probability of making a Type I error, which means rejecting the null hypothesis when it is actually true.
2. The Type II error, denoted by \( \beta \), is the probability of failing to reject the null hypothesis when the alternative hypothesis is true.
3. Understand that \( \alpha \) and \( \beta \) are inversely related because decreasing \( \alpha \) makes the rejection region smaller, which means it becomes harder to reject the null hypothesis.
4. As a result, when \( \alpha \) decreases, the probability of failing to reject the null hypothesis when it is false (i.e., making a Type II error, \( \beta \)) tends to increase.
5. This trade-off occurs because tightening the criteria to avoid Type I errors (lower \( \alpha \)) generally increases the chance of missing a true effect, thus increasing \( \beta \).

Key Concepts

Type I Error (α)

A Type I error occurs when the null hypothesis is incorrectly rejected, meaning a false positive. The level of significance, α, represents the probability of making this error and is set by the researcher to control the risk of false positives.

Type II Error (β)

A Type II error happens when the null hypothesis is not rejected even though it is false, resulting in a false negative. The probability of this error, β, depends on factors like sample size, effect size, and the chosen significance level.

Trade-off Between α and β

Decreasing α (making the test more stringent) reduces the chance of a Type I error but generally increases β, the chance of a Type II error. This trade-off occurs because stricter criteria make it harder to detect true effects, thus increasing the risk of missing them.