42 9.2 Outcomes, 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:
| H0 is actually | ||
|---|---|---|
| Action | True | False |
| Do not reject H0 | Correct Outcome | Type II Error ([latex]\beta[/latex]) |
| Reject H0 | Type I Error ([latex]\alpha[/latex]) | Correct Outcome |
The four possible outcomes in the table are:
- The decision is not to reject H0 when H0 is true (correct decision).
- The decision is to reject H0 when H0 is true (incorrect decision known as a Type I error).
- The decision is not to reject H0 when H0 is false (incorrect decision known as a Type II error).
- The decision is to reject H0 when H0 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 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.
β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.
α and β should be as small as possible because they are probabilities of errors. They are rarely zero.
The Power of the Test is 1 – β.
Since β is probability of making type II error, we want this probability to be small.
In other words, we want the value 1 – β to be as closed to one as possible.
Increasing the sample size can increase the Power of the Test.
Example 1
Suppose the null hypothesis, H0, 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.
α = Probability that Frank thinks his rock climbing equipment may not be safe when it really is safe.
β = Probability that Frank thinks his rock climbing equipment may be safe when it is not safe.
| Null Hypothesis: The rock climbing equipment is safe. | ||
| Frank’s decision | True (The equipment is safe) | False (The equipment is not safe.) |
| Not reject H0 | Correct decision | Type II Error |
| Reject H0 | Type I Error | Correct decision |
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.)
Try It
Suppose the null hypothesis, H0, is: the blood cultures contain no traces of pathogen X.
State the Type I and Type II errors.
[practice-area rows=”2″][/practice-area]
Solution
Type I error: The researcher thinks the blood cultures do contain traces of pathogen X, when in fact, they do not.
Type II error: The researcher thinks the blood cultures do not contain traces of pathogen X, when in fact, they do.
| Null h