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Outcomes and the Type I and Type II Errors

When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H₀ and the decision to reject or not. The outcomes are summarized in the following table:

table 9.2

The four possible outcomes in the table are:

The decision is cannot reject H₀ when H₀ is true (correct decision).
The decision is cannot accept H₀ when H₀ is true (incorrect decision known as a Type I error). This case is described as "rejecting a good null". As we will see later, it is this type of error that we will guard against by setting the probability of making such an error. The goal is to NOT take an action that is an error.
The decision is cannot reject H₀ when, in fact, H₀ is false (incorrect decision known as a Type II error). This is called "accepting a false null". In this situation you have allowed the status quo to remain in force when it should be overturned. As we will see, the null hypothesis has the advantage in competition with the alternative.
The decision is cannot accept H₀ when H₀ is false (correct decision).

Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.

α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true: rejecting a good null.
β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. (1 − β) is called the Power of the Test.

α and β should be as small as possible because they are probabilities of errors.

Statistics allows us to set the probability that we are making a Type I error. The probability of making a Type I error is α. Recall that the confidence intervals in the last unit were set by choosing a value called Z_α (or t_α) and the alpha value determined the confidence level of the estimate because it was the probability of the interval failing to capture the true mean (or proportion parameter p). This alpha and that one are the same.

Example 9.4

Source: Introductory to Business Statistics by OpenStax, Retrieved from https://openstax.org

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