## 3.3 Type I and Type II Errors

Whenever we carry out a hypothesis test, there is always a chance we will arrive at the incorrect conclusion. For example, we may reject $$H_0$$ when $$H_0$$ was actually true. Or, we may fail to reject $$H_0$$ when $$H_0$$ was actually false. We can summarise these types of errors as follows:

There are two types of error that can occur:

1. Type I error: Reject $$H_0$$ when $$H_0$$ is true.
2. Type II error: Fail to reject $$H_0$$ when $$H_0$$ is false.

Consider the level of significance, $$\alpha$$. If we have that $$\alpha = 0.05$$, this means that, assuming $$H_0$$ is true, as long as there is less than a 5% chance of us obtaining the sample mean we did, we will reject $$H_0$$. This means that there is actually a 5% chance we will end up rejecting $$H_0$$ when it was actually true. This leads us to the following fact:

Probability of Type I error:

The probability of making a Type I error is equal to the significance level, $$\alpha$$.

The researcher carrying out the test controls the level of significance. So, they can feasibly choose a smaller $$\alpha$$ in order to reduce the risk of making a Type I error. However, in doing so, it would become harder to reject $$H_0$$ when $$H_0$$ was actually false. So there is a trade-off between Type I and Type II errors. As mentioned earlier, $$\alpha = 0.05$$ is most commonly chosen because many believe this is a small enough risk of making a Type I error while not making the chance of a Type II too great.