Answer to Question #212798 in Statistics and Probability for Jhay

Question

How do you determine the appropriate rejection region and what is its significance in hypothesis testing?

Answer 1

A rejection region is an area of a graph where you would reject the null hypothesis (assuming your test results fall into that area).

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:

1. Be repeatable.
2. 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.

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.

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 region, you reject the null hypothesis.