9.2 Hypothesis testing of single mean and single proportion: outcomes  (Page 91/3)

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis $H_o$ and the decision to reject or not. The outcomes are summarized in the following table:

The four possible outcomes in the table are:

• The decision is to not reject $H_o$ when, in fact, $H_o$ is true (correct decision).
• The decision is to reject $H_o$ when, in fact, $H_o$ is true (incorrect decision known as a Type I error ).
• The decision is to not reject $H_o$ when, in fact, $H_o$ is false (incorrect decision known as a Type II error ).
• The decision is to reject $H_o$ when, in fact, $H_o$ is false ( correct decision whose probability is called the Power of the Test ).

Each of the errors occurs with a particular probability. The Greek letters $\alpha$ and $\beta$ represent the probabilities.

$\alpha$ = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

$\beta$ = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

$\alpha$ and $\beta$ should be as small as possible because they are probabilities of errors. They are rarely 0.

The Power of the Test is $1-\beta$ . Ideally, we want a high power that is as close to 1 as possible. Increasing the sample size can increase the Power of the Test.

The following are examples of Type I and Type II errors.