Type-I and Type-II Errors and the Power of a Test for AP Statistics
Practice problems for these concepts can be found at:
- Confidence Intervals and Introduction to Inference Multiple Choice Practice Problems for AP Statistics
- Confidence Intervals and Introduction to Inference Free Response Practice Problems for AP Statistics
- Confidence Intervals and Introduction to Inference Review Problems for AP Statistics
- Confidence Intervals and Introduction to Inference Rapid Review for AP Statistics
When we do a hypothesis test, we never really know if we have made the correct decision or not. We can try to minimize our chances of being wrong, but there are trade-offs involved. If we are given a hypothesis, it may be true or itmay be false. We can decide to reject the hypothesis or not to reject it. This leads to four possible outcomes:
Two of the cells in the table are errors and two are not. Filling those in, we have
The errors have names that are rather unspectacular: If the (null) hypothesis is true, and we mistakenly reject it, it is a Type-I error. If the hypothesis is false, and we mistakenly fail to reject it, it is a Type-II error. We note that the probability of a Type-I error is equal to α, the significance level (this is because a P-value < α causes us to reject H0. If H0 is true, and we still decide to reject it, we have made a Type-I error). We call probability of a Type-II error β. Filling in the table with this information, we have:
The cell in the lower right-hand corner is important. An honest person does not want to foist a false hypothesis on the public and hopes that a study would lead to a correct decision to reject it. The probability of correctly rejecting a false hypothesis (in favor of the alternative) is called the power of the test. The power of the test equals 1 – β. Finally, then, our decision table looks like this:
example: Sticky Fingers is arrested for shoplifting. The judge, in her instructions to the jury, says that Sticky is innocent until proved guilty. That is, the jury's hypothesis is that Sticky is innocent. Identify Type-I and Type-II errors in this situation and explain the consequence of each.
solution: Our hypothesis is that Sticky is innocent. A Type-I error involves mistakenly rejecting a true hypothesis. In this case, Sticky is innocent, but because we reject innocence, he is found guilty. The risk in a Type-I error is that Sticky goes to jail for a crime he didn't commit.
A Type-II error involves failing to reject a false hypothesis. If the hypothesis is false, then Sticky is guilty, but because we think he's innocent, we acquit him. The risk in Type-II error is that Sticky goes free even though he is guilty.
In life we often have to choose between possible errors. In the example above, the choice was between sending an innocent person to jail (a Type-I error) or setting a criminal free (a Type-II error). Which of these is the most serious error is not a statistical question—it's a social one.
We can decrease the chance of Type-I error by adjusting α. By making α very small, we could virtually ensure that we would never mistakenly reject a true hypothesis. However, this would result in a large Type-II error because we are making it hard to reject the null under any circumstance, even when it is false.
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