What are Type I and Type II errors in hypothesis testing?

Type I error occurs when a true null hypothesis is rejected, while Type II error happens when a false null hypothesis is not rejected.

In hypothesis testing, we start with a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis is a statement of no effect or no difference and the alternative hypothesis is what we are trying to prove. When we conduct a test, we either reject the null hypothesis in favour of the alternative, or we do not reject the null hypothesis.

A Type I error, also known as a false positive, occurs when we incorrectly reject a true null hypothesis. This means we are seeing an effect or difference when in reality there isn't one. For example, if we are testing a new drug and we conclude that it works when it actually doesn't, we have made a Type I error. The probability of making a Type I error is denoted by the Greek letter alpha (α), which is also the significance level of the test. The lower the value of α, the lower the chance of making a Type I error.

On the other hand, a Type II error, also known as a false negative, happens when we fail to reject a false null hypothesis. This means we are not seeing an effect or difference when in reality there is one. For instance, if we conclude that the new drug doesn't work when it actually does, we have made a Type II error. The probability of making a Type II error is denoted by the Greek letter beta (β). The power of a test, which is the probability of correctly rejecting a false null hypothesis, is 1 - β. Therefore, the higher the power of a test, the lower the chance of making a Type II error.

In summary, Type I and Type II errors represent two types of incorrect conclusions that can be made in hypothesis testing. The risk of these errors can be controlled to some extent by choosing appropriate significance levels and ensuring the test has sufficient power. However, it's important to remember that there is always a trade-off between these two types of errors. Reducing the risk of one type of error increases the risk of the other.