Hypothesis Testing and Type I and Type II Error: The Murder Trial

Source: http://i.stack.imgur.com/FPCq0.jpg

It has been a while since I last posted an article on this blog. Let's take a break from Economics and take a look at the beauty of Statistics. To be honest, I found Statistics to be very boring during my undergrad. That said, now I'm very intrigued of the power of Statistics. Statistics is more than just about calculating the tedious probability. I now find that probability is a useful foundation in Statistics, but there is more to explore in Statistics. To claim that Statistics is all about probability is like to claim that Economics is all about demand and supply. Anyway, let's get to our topic for today. In Statistics you may have heard about hypothesis testing and the Type 1 and Type 2 error. My Statistics Professor explained this concept in a simple way by using a murder trial as an analogy. In this article, I hope you will have tons of fun learning the concept of hypothesis testing, Type 1 and Type 2 Error. I will keep our discussion simple without getting into any mathematical formula. I hope I can further discuss a real statistics example in the next article.

Hypothesis Testing

Okay, let’s say there is a homicide case in our community and the police arrest a man called Suspect A. We, as a young economist, was hired for no reason to find out whether Suspect A is a murderer. Probably it is because economists love to come up with a statement and love to test whether the statement is correct or false. Therefore, we need to gather evidence and evaluate whether there is sufficient evidence to prove that Suspect A is guilty as charge. This process of evaluating a hypothesized statement based on the evidence is called Hypothesis Testing in Statistics.  

First, we should begin by stating our null hypothesis. A null hypothesis is usually (but not always) the hypothesis which one wants to reject or nullify. In the legal system, generally a person is innocent until proven guilty. Thus, in our murder case, our null hypothesis is that the suspect is innocent. In other words, the suspect did not commit the murder. On the other hand, the alternative hypothesis is the opposite of our null hypothesis and we are trying to find evidence to prove that the alternate hypothesis is correct and reject the null hypothesis. Simply put, the alternative hypothesis in our case is that the suspect killed his wife, which means the man is guilty. Here is the summary so far:

Null Hypothesis: H0: Suspect A is innocent

Alternative Hypothesis: Ha: Suspect A is guilty

Type I and Type II Error

In evaluating the validity and accuracy of our hypothesis, we may unintentionally make two types of mistakes or errors. First, based on our evidence, we may come to a conclusion that the suspect is guilty and sentence an “innocent” person to jail. For this decision, we make a serious mistake and send the poor man to jail for the crime that he did not commit. This is called “Type I Error” in Statistics. In a technical explanation, Type I Error (denoted by alpha) is the probability of rejecting a true hypothesis.

Another mistake that we may make in this murder case trial is that we may not find enough evidence to prove that he is a murderer and acquit (free) the man who in fact is the murder. In this situation, we set free a guilty person. This is known as “Type II Error”. In a formal way, Type II Error is the probability of accepting a false hypothesis.

In summary,

Ruling	In Reality
	The man is innocent	The man is guilty
Sentence the man to jail	Type I Error	Correct Decision
Set the man free	Correct Decision	Type II Error

In general,

Decision	In Reality
	Null hypothesis is true	Null hypothesis is false
Reject null hypothesis	Type I Error	Correct Decision
Accept null hypothesis	Correct Decision	Type II Error

In both of these circumstances, sending an innocent man to prison and setting free a guilty man are both wrong decisions that we want to avoid. I also sum up what we have discussed so far into a simple table. The ideal scenario is to minimize the two types of error. However, we will later learn that Type I and Type II Error are inversely correlated. Hope you are hooked. Ok this is it for today. I hope you now have some intuitive understanding of hypothesis testing, Type I and Type II Error. Next time, I will try to go a bit deeper into the formula to conduct hypothesis testing.