Type I & Type II Errors (College Board AP® Statistics): Revision Note
Type I & Type II errors
What are the four possible conclusions of a hypothesis test?
There are four possible conclusions of a hypothesis test
Two good outcomes
H0 was false and the test rejected H0
H0 was true and the test failed to reject H0
Two bad outcomes (errors)
H0 was true and the test rejected H0 (a Type I error)
H0 was false and the test failed to reject H0 (a Type II error)
Conclusion | |||
|---|---|---|---|
Rejected | Failed to reject | ||
Reality |
| Type I | No error |
| No error | Type II | |
What is a Type I error?
Type I errors occur when a hypothesis test gives convincing statistical evidence to reject H0 despite it being true in reality
This is sometimes called a “false positive”
In a court case, this would be when the defendant is found guilty despite being innocent
What is a Type II error?
Type II errors are when a hypothesis test does not give convincing statistical evidence to reject H0 despite it being false in reality
This is sometimes called a “false negative”
In a court case, this would be when the defendant is found innocent despite being guilty
Worked Example
A city council is debating whether to fund a new town-wide recycling initiative. The council will fund the initiative only if they find convincing statistical evidence that more than 60 percent of residents support it. A random sample of residents is surveyed, and a hypothesis test is conducted. The resulting p-value is 0.12. Using a significance level of α=0.05, the council decides not to fund the initiative. However, a complete census later reveals that 68 percent of all residents actually supported the initiative.
Which of the following statements best describes the statistical error that was made in this study?
(A) A Type I error, because there was convincing statistical evidence that the alternative hypothesis was true, but it was not.
(B) A Type I error, because there was not convincing statistical evidence that the alternative hypothesis was true, but it was.
(C) A Type II error, because there was convincing statistical evidence that the alternative hypothesis was true, but it was not.
(D) A Type II error, because there was not convincing statistical evidence that the alternative hypothesis was true, but it was.
Answer:
Based on the prompt, the null hypothesis (
) is that the proportion of residents who support the initiative is p=0.60 (or p≤0.60), and the alternative hypothesis (
) is p>0.60
Because the p-value (0.12) is greater than the α level (0.05), the council correctly follows procedure and fails to reject the null hypothesis
Failing to reject the null hypothesis means there is not convincing statistical evidence to support the alternative hypothesis
However, the census reveals that the true population proportion is 0.68, meaning the alternative hypothesis (p>0.60) is actually true
A Type II error occurs when there is not convincing statistical evidence that the alternative hypothesis is true (due to a large p-value), but it actually is true
Therefore, the correct answer is D
Why the distractors are incorrect:
(A) correctly states the definition of a Type I error, but misapplies it to the scenario
The council did not find convincing evidence for the alternative hypothesis
(B) improperly mixes the label of a Type I error with the definition of a Type II error
(C) improperly mixes the label of a Type II error with the definition of a Type I error
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