A Chi-Square Crime (Part 1)

The following is an excerpt from the free, on-line lab manual used in the introductory cell biology course at Cal Poly (see the Resources section).  This story was inspired by Martina Bremer, a Statistician at San Jose State University (She's not a a watch thief!  She introduced me to the idea of the chi-square test being like a trial.)  I think this example is effective in helping students think about the null hypothesis and the concept of significance.

The Case of the Fancy Watch

"Innocent until proven guilty" is a foundation of our legal system. A courtroom can actually be a good analogy for scientific decision making. Before a person can be convicted significant evidence against that person must be presented. In a courtroom there is a null hypothesis:

null hypothesis (H₀) = The suspect is innocent.

The prosecutor will present evidence against the null hypothesis (i.e. trying to show that the suspect is guilty). The jury will decide if the evidence against the suspect is significant.

Let's say a robbery has taken place at a fancy store…a jewel-encrusted Piaget watch has been stolen! Soon after, a suspect is arrested and brought to court. The evidence against him is simple; when the police found him he was WEARING a jewel-encrusted Piaget watch! The jury must consider this question…What is the chance that this person is innocent (the null hypothesis is true) given that he was wearing a jewel-encrusted Piaget watch?

image source: piaget.com

Let's say that only 0.001% (1 in 1000 people) in the town wear jewel-encrusted Piaget watches on a regular basis. If that were the case, the evidence against the suspect would be significant. Looking at it another way…the chance that he is innocent and wearing jewel-encrusted Piaget watch is just 1 in 1000 or (0.001). The jury would probably find him guilty (i.e. they would REJECT the null hypothesis.)

But what if, in this particular town, 75% of people were walking around wearing jewel-encrusted Piaget watches. Now the case against the suspect isn't very strong. The fact that he was found wearing a jewel-encrusted Piaget watch is nonsignificant. Saying it another way…there is a 75% chance that the suspect is innocent and wearing a Jewel-encrusted Piaget watch just by coincidence! The jury would most likely NOT reject the null hypothesis and find the suspect not guilty.

Put yourself on that jury…at what point would you be comfortable rejecting the null hypothesis and convicting the suspect? What if 10% of people in the town wore jewel-encrusted Piaget watches? What about 1%? At what point would the evidence (the suspect was wearing a watch) become significant?

In a scientific study, the cut off for significance is 5% (in our courtroom, the jury would reject the null hypothesis and convict the suspect if the chance that he is innocent and wearing a fancy watch is 0.05 or lower.)

In part 2 of this post I'll provide some examples I like for introducing the chi-square calculations.  Thanks for reading!