## A Case of Restaurant Arson and the Reasoning of Statistical Inference

I know of a case of a restaruant owner who was convicted for burning down his restaurant for the purpose of collecting insurance money. It turns out that this case is a good example for introducing the reasoning process of statistical inference from an intuitive point of view.

This restaurant owner was convicted for burning down his restaurant and received a lengthy jail sentence. What was the red flag that alerted the insurance company that something was not quite right in the first place? The same restaurant owner’s previous restaurant was also burned to the ground!

Of course, the insurance company could not just file charges against the owner simply because of two losses in a row. But the two losses in a row did raise a giant red flag for the insurer, which brought all its investigative resource to bear on this case. But for us independent observers, this case provides an excellent illustration of an intuitive reasoning process for statistical inference.

The reasoning process behind the suspicion is a natural one for many people, including students in my introductory statistics class. Once we learn that there were two burned down restaurants in a row, we ask: are the two successive losses just due to bad luck or are the losses due to other causes such as fraud? Most people would feel that two successive fire losses in a row are unlikely if there is no fraud involved. It is natural that we would settle on the possibility of fraud rather than attributing the losses to bad luck.

The reasoning process is mostly unspoken and intuitive. For the sake of furthering the discussion, let me write it out:

1. The starting point of the reasoning process is an unspoken belief that the restaurant owner did not cause the fire (we call this the null hypothesis).
2. We then evaluate the observed data (losing two restaurants in a row to fire). What would be the likelihood of this happening if the null hypothesis were true? Though we assess this likelihood intuitively, this probability is called a p-value, which we feel is small in this case.
3. Since we feel that the p-value is very small, we reject the initial belief (null hypothesis), rather than believing that the rare event of two fire losses in a row was solely due to chance.

The statistical inference problems that we encounter in a statistics course are all based on the same intuitive reasoning process described above. However, unlike the restaurant arson case described here, the reasoning process in many statistical inference problems does not come naturally for students in introductory statistics classes. The reason may be that it is not easy to grasp intuitively the implication of having a small p-value in many statistical inference problems. It is difficult for some students to grasp why a small p-value should lead to the rejection of the null hypothesis.

In our restaurant arson example, we do not have to calculate a p-value and simply rely on our intuition to know that the p-value, whatever it is, is likely to be very small. In statistical inference problems that we do in a statistics class, we need a probability model to calculate a p-value. Beginning inference problems in an introductory class usually are based on normal distributons or in some cases the binomial distribution. With the normal model or binomial model, our intuition is much less reliable in computing and interpreting the p-value (the probability of obtaining the observed data given the null hypothesis is true). This is a challenge for students to overcome. To address this challenge, it is critical that students have a good grounding of normal distributions and binomial distribution. Simulation projects can also help students gain an intuitive sense of p-value.

In any case, just know that in terms of reasoning, any inference problem would work the same way as the intuitive example described here. It would start off with a question or a claim. The solution would start with a null hypothesis, which is a neutral proposition (e.g. the restaurant owner did not do it). We then evaluate the observed data to calculate the p-value. The p-value would form the basis for judging the strength of the observed data: the smaller the p-value, the less credible the null hypothesis and the stronger the case for rejecting the null hypothesis.

Repeated large losses for the same claimant raise suspicion for property insurance companies as well as life insurance companies. The concept of p-value is important for insurance companies. The example of repeated large insurance losses described here is an excellent example for illustrating the intuitive reasoning behind statistical inference.