Thought and Knowledge by Halpern Diane F

Author:Halpern, Diane F.
Language: eng
Format: epub
ISBN: 978-1-134-63793-5
Publisher: Taylor and Francis

Thinking about Errors

To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at changing his mind.

—Wendell Johnson, p. 39 (1946)

When we try to understand relationships by devising and testing hypotheses, we will sometimes be wrong. This idea is expanded on more fully in the next chapter, which concerns understanding probabilities. For now, consider this possibility: Suppose that you drive into work every day with a friend. Every morning you stop at a drive-through window and buy coffee. You decide that, instead of hassling every morning with who will pay (“I’ll get it—No, no let me”), he will flip a coin. When the outcome is heads, he will pay; when the outcome is tails, you will pay. Sounds fair enough, but on nine of the last ten days, the coin landed with tails up. Do you think that your friend is cheating?

The truth is that your friend is either cheating or he is not cheating. Unfortunately, you don’t know which is true. Nevertheless, you need to make a decision. You will decide either that he is cheating or he is not cheating. Thus, there are four possibilities: (1) He is cheating, and you correctly decide that he is cheating; (2) He is not cheating, and you correctly decide that he is not cheating; (3) He is cheating, and you incorrectly decide that he is not cheating; and (4) He is not cheating, and you incorrectly decide that he is cheating. With these four possibilities, there are two ways that you can be right and two ways that you can be wrong. These four combinations are shown in Table 6.1.

As you can see from Table 6.1, there are two different ways that we can make errors in any hypothesis-testing situation. These two different errors are not equally “bad.” It is far worse to decide that your friend is cheating when he is not (especially if you accuse him of cheating) than it is to decide that he is not cheating when he is. Thus, you would want stronger evidence to decide that he is cheating than you would want to decide that he is not cheating. In other words, you need to consider the relative “badness” of different errors when testing hypotheses.

If you take a course in statistics or experimental design, you’ll find that the idea of error “badness” is handled by requiring different levels of confidence for different decisions. The need to consider different types of errors is found in many contexts. A basic principle of our legal system is that we have to be very certain that someone has committed a crime (beyond a reasonable doubt) before we can convict her. By contrast, we do not have to be convinced beyond a reasonable doubt that she is innocent because wrongly deciding that someone is innocent is considered a less severe error than wrongly deciding that someone is guilty. Similarly, when you are

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