Hypothesis Testing > Rejection Region

## What is a Rejection Region?

The main purpose of statistics is to test theories or results from experiments. For example, you might have invented a new fertilizer that you think makes plants grow 50% faster. In order to prove your theory is true, your experiment must:

- Be repeatable.
- Be compared to a known fact about plants (in this example, probably the average growth rate of plants
*without*the fertilizer).

We call this type of statistical testing a hypothesis test. **The rejection region (also called a critical region) is a part of the testing process. **Specifically, it is an area of probability that tells you if your theory (your “”hypothesis”) is probably true.

## Rejection Regions and Probability Distributions

Every rejection region can be drawn on a probability distribution. The image above shows a t-distribution with a two-tailed rejection region. It’s also possible to have a rejection region in one tail only.

## Two Tailed vs One Tailed Rejection Regions

Which type of test is determined by your null hypothesis statement. For example, if your statement asks *“Is the average growth rate greater than 10cm a day?”* that’s a one tailed test, because you are only interested in one direction (greater than 10cm a day). You could also have a single rejection region for “less than”. For example, “Is the growth rate less than 10cm a day?” A two tailed test, with two rejection regions, would be used when you want to know if there’s a difference in both directions (greater than **and** less than).

## Rejection Regions and Alpha Levels

You, as a researcher, choose the alpha level you are willing to accept. For example, if you wanted to be 95% confident that your results are significant, you would choose a 5% alpha level (100% – 95%). That 5% level is the **rejection region**. For a one tailed test, the 5% would be in one tail. For a two tailed test, the rejection region would be in two tails.

## Rejection Regions and P-Values.

There are two ways you can test a hypothesis: with a p-value and with a critical value.

**P-value method**: When you run a hypothesis test (for example, a z test), the result of that test will be a p value. The p value is a “probability value.” It’s what tells you if your hypothesis statement is probably true or not. If the value falls in the rejection region, it means you have statistically significant results; You can reject the null hypothesis. If the p-value falls outside the rejection region, it means your results aren’t enough to throw out the null hypothesis. What is **statistically significant**? In the example of the plant fertilizer, a statistically significant result would be one that shows the fertilizer does indeed make plants grow faster (compared to other fertilizers).

**Rejection Region method with a critical value**: The steps are exactly the same. However, instead of calculating a p-value you calculate a critical value. If the value falls inside the rejection region, you reject the null hypothesis.

**Next**: What is an Acceptance Region?

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