What is the difference between Type I Error & Type II Error? Also, Explain the Power of the test?

When we perform hypothesis testing we consider two types of Error, Type I error and Type II error, sometimes we reject the null hypothesis when we should not or choose not to reject the null hypothesis when we should.

A Type I Error is committed when we reject the null hypothesis when the null hypothesis is actually true. On the other hand, a Type II error is made when we do not reject the null hypothesis and the null hypothesis is actually false.

The probability of a Type I error is denoted by α and the probability of Type II error is denoted by β.

For a given sample n, a decrease in α will increase β and vice versa. Both α and β decrease as n increases.

The table given below explains the situation around the Type I error and Type II error:

Decision Null Hypothesis is true Null hypothesis is false
Reject the Null Hypothesis Type I error Correct Decision
Fail to reject Null Hypothesis Correct Decision Type II error

Two correct decisions are possible: not rejecting the null hypothesis when the null hypothesis is true and rejecting the null hypothesis when the null hypothesis is false.

Conversely, two incorrect decisions are also possible: Rejecting the null hypothesis when the null hypothesis is true(Type I error), and not rejecting the null hypothesis when the null hypothesis is false (Type II error).

Type I error is false positive while Type II error is a false negative.

Power of Test: The Power of the test is defined as the probability of rejecting the null hypothesis when the null hypothesis is false. Since β is the probability of a Type II error, the power of the test is defined as 1- β. In advanced statistics, we compare various types of tests based on their size and power, where the size denotes the actual proportion of rejections when the null is true and the power denotes the actual proportion of rejections when the null is false.