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Tuesday, May 11, 2010

Which test would you choose?

Suppose that today you have a test about a book that you were supposed to read. The book has 100 chapters, but you only had time to read 75 of them. (You were too busy watching How I Met Your Mother, making lists, driving to Sonic, sleeping, or blogging to finish the rest of the chapters. We don't really have time to debate about whether or not these activities were the best use of your time.) You walk into the test very nervous (because you only know 75% of what you're supposed to know), and your teacher (drumroll) has two stacks of DIFFERENT tests. She says she is going to let each student choose which test they want to take. She tells you your options and you analyze them.

Option #1 is a test with 100 fill-in-the-blank questions, one question about each chapter. Since you read 75% of the chapters, are adequately intelligent, and the answers are obvious if you read the chapters they are about, this option will guarantee you a grade of 75 on the test. So your expected grade if you choose this option is:
EV(option #1) = .75 x 100 = 75

Option #2 is a test with only one question, about a random chapter. Since you read 75% of the chapters, are adequately intelligent, and the answers are obvious if you read the chapters they are about, with this option you have a 75% chance of getting a 100 on the test, but a 25% chance that you will get a 0. So your expected grade if you choose this option is:
EV(option #2) = .75 x 100 + .25 x 0 = 75

Either way, your expected grade is a 75, so you should be indifferent about which option you choose. 

Before continuing with the conclusion of this post, I would like to make a quick note about an objection to my model that an adequately intelligent reader may have. I was thinking that the model might be different for a multiple choice test because if you took the long test, then you would have a 25% chance of answering correctly each of the 25 questions that you didn't know, thus raising your expected grade for the long test to 81.25. But, the expected value for the short test would also increase because if the single question was about a chapter you didn't read, you would have a 25% chance of getting it right. In the case of a multiple-choice test, the expected values of both options would be 81.25 (still equal).

However, there are other factors to consider.

First, what is your level of risk-aversion? A very risk-averse individual would prefer a guaranteed 75. But a risk-lover would hope for 100 even if it means she might get a 0. I will refer to this as the risk-aversion effect.

Next, how much do you value the time you will waste taking the test? If each question takes one minute to complete, the short test will only take one minute while the long test will take 100 minutes to complete. I will refer to this as the time-wasting effect.

A stereotypical extreme b.a. (badas*, not necessarily bachelor of arts) is a risk-lover and  thinks tests are a huge waste of time (she would rather be riding her motorcycle), so would choose the short test.

A stereotypical extreme dork is risk-averse and enjoys taking tests (she learns for fun so would probably use spare time to read anyway), and would thus choose the long test.

For a more moderate student that is neither of the two extremes, both the risk-aversion effect and the time-wasting effect will be present, so one must look at her individual personality to see which effect overpowers the other.

I myself am a risk-averse individual (I wouldn't gamble high-stakes and uncertainty stresses me out), so the risk-aversion effect is present. However, I am also an economist, and the opportunity cost of those 99 extra minutes wasted taking the long test could have been put to much better use (such as riding a motorcycle or reading the last 25 chapters), so the time-wasting effect is also present.

So the real question isn't "Which test would you choose?" but rather "What is your individual personality?"


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