Chapter 8: Hypothesis Testing with One Sample
8.2 Outcomes and the Type I and Type II Errors
Learning Objectives
By the end of this section, the student should be able to:
- Differentiate between Type I and Type II errors in a hypothesis test.
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 [latex]H_{0}[/latex] and the decision to reject or not. The outcomes are summarized in the following table:
| ACTION | [latex]H_{0}[/latex] IS ACTUALLY | … |
|---|---|---|
| True | False | |
| Do not reject[latex]H_{0}[/latex] | Correct Outcome | Type II error |
| Reject [latex]H_{0}[/latex] | Type I Error | Correct Outcome |
The four possible outcomes in the table are:
Each of the errors occurs with a particular probability. The Greek letters [latex]\alpha[/latex] and [latex]\beta[/latex] represent the probabilities.
[latex]\alpha[/latex] = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.
[latex]\beta[/latex] = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.
[latex]\alpha[/latex] and [latex]\beta[/latex] should be as small as possible because they are probabilities of errors. They are rarely zero.
The Power of the Test is [latex]1 - \beta[/latex]. Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test.
The following are examples of Type I and Type II errors.
Suppose the null hypothesis, [latex]H_{0}[/latex], is: Frank’s rock climbing equipment is safe.
Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.
[latex]\alpha[/latex] = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. [latex]\beta[/latex] = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.
Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)
Example
Suppose the null hypothesis, [latex]H_{0}[/latex], is: the blood cultures contain no traces of pathogen [latex]X[/latex]. State the Type I and Type II errors.
Solution
Type I error: The researcher thinks the blood cultures do contain traces of pathogen [latex]X[/latex], when in fact, they do not.
Type II error: The researcher thinks the blood cultures do not contain traces of pathogen [latex]X[/latex], when in fact, they do.
Your Turn!
Suppose the null hypothesis, [latex]H_{0}[/latex], is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.
Type I error: The emergency crew thin