How is the p-value of a hypothesis test related to type I and type II error?
P-value is the probability of a type I error.
The p-value of a hypothesis test is related to type I and type II errors through the concept of statistical significance.
Type I error occurs when we reject a null hypothesis that is actually true. In other words, we falsely conclude that there is a significant difference or effect when there isn't one. This is also known as a false positive.
Type II error occurs when we fail to reject a null hypothesis that is actually false. In other words, we fail to detect a difference or effect that actually exists. This is also known as a false negative.
The p-value is a measure of the strength of evidence against the null hypothesis. It represents the probability of observing a test statistic as extreme as the one obtained if the null hypothesis were true.
A smaller p-value indicates stronger evidence against the null hypothesis. If the p-value is below a pre-determined significance level (typically 0.05 or 0.01), we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis.
When we set a lower significance level (e.g., 0.01), we are more cautious in rejecting the null hypothesis. This reduces the likelihood of Type I error (false positive) but increases the likelihood of Type II error (false negative). Conversely, if we set a higher significance level (e.g., 0.10), we are more likely to reject the null hypothesis, which increases the risk of Type I error (false positive) but decreases the risk of Type II error (false negative).
In summary, the p-value is used to determine whether the observed data provides enough evidence to reject the null hypothesis. The choice of significance level (α) determines the trade-off between Type I and Type II errors - a lower significance level reduces Type I error but increases Type II error, while a higher significance level increases Type I error but reduces Type II error.