Hypothesis testing is a topic at the heart of statistics. This technique belongs to a realm known as inferential statistics. Researchers from all sorts of different areas, such as psychology, marketing, and medicine, formulate hypotheses or claims about a population being studied. The ultimate goal of the research is to determine the validity of these claims. Carefully designed statistical experiments obtain sample data from the population. The data is in turn used to test the accuracy of a hypothesis concerning a population.

### The Rare Event Rule

Hypothesis tests are based upon the field of mathematics known as probability. Probability gives us a way to quantify how likely it is for an event to occur. The underlying assumption for all inferential statistics deals with rare events, which is why probability is used so extensively. The rare event rule states that if an assumption is made and the probability of a certain observed event is very small, then the assumption is most likely incorrect.

The basic idea here is that we test a claim by distinguishing between two different things:

- An event that easily occurs by chance.
- An event that is highly unlikely to occur by chance.

If a highly unlikely event occurs, then we explain this by stating that a rare event really did take place, or that the assumption we started with was not true.

### Prognosticators and Probability

As an example to intuitively grasp the ideas behind hypothesis testing, we’ll consider the following story.

It’s a beautiful day outside so you decided to go on a walk. While you are walking you are confronted by a mysterious stranger. “Do not be alarmed,” he says, “this is your lucky day. I am a seer of seers and a prognosticator of prognosticators. I can predict the future, and do it with greater accuracy than anyone else. In fact, 95% of the time I’m right. For a mere $1000, I will give you the winning lottery ticket numbers for the next ten weeks. You‘ll be almost sure of winning once, and probably several times.”

This sounds too good to be true, but you are intrigued. “Prove it,” you reply. “Show me that you really can predict the future, then I’ll consider your offer.”

“Of course. I can‘t give you any winning lottery numbers for free though. But I will show you my powers as follows. In this sealed envelope is a sheet of paper numbered 1 through 100, with 'heads' or 'tails' written after each of them. When you go home, flip a coin 100 times and record the results in the order that you get them. Then open the envelope and compare the two lists. My list will accurately match at least 95 of your coin tosses.”

You take the envelope with a skeptical look. “I will be here tomorrow at this same time if you decide to take me up on my offer.”

As you walk back home, you assume that the stranger has thought up a creative way to con people out of their money. Nevertheless, when you get back home, you flip a coin and write down which tosses give you heads, and which ones are tails. Then you open the envelope and compare the two lists.

If the lists only match in 49 places, you would conclude that the stranger is at best deluded and at worse conducting some sort of scam. After all, chance alone would result in being correct about one half of the time. If this is the case, you would probably change your walking route for a few weeks.

On the other hand, what if the lists matched 96 times? The likelihood of this occurring by chance is extremely small. Due to the fact that predicting 96 of 100 coin tosses is exceptionally improbable, you conclude that your assumption about the stranger was incorrect and he can indeed predict the future.

### The Formal Procedure

This example illustrates the idea behind hypothesis testing and is a good introduction to further study. The exact procedure requires specialized terminology and a step by step procedure, but the thinking is the same. The rare event rule provides the ammunition to reject one hypothesis and accept an alternate one.