Hypothesis Testing with Z-Test: Significance Level and Rejection Region
If you want to understand why hypothesis testing works, you should first have an idea about the significance level and the reject region. We assume you already know what a hypothesis is, so let’s jump right into the action.However, as with any test, there is a small chance that we could get it wrong and reject a null hypothesis that is true.So, the probability of making this error. Typical values for α are 0.01, 0.05 and 0.1. It is a value that we select based on the certainty we need. In most cases, the choice of α is determined by the context we are operating in, but 0.05 is the most commonly used value.For instance, if we want to predict how much Coca Cola its consumers drink on average, the difference between 12 ounces and 12.1 ounces will not be that crucial. So, we can choose a higher significance level like 0.05 or 0.1.The university dean believes that on average students have a GPA of 70%. Being the data-driven researcher that you are, you can’t simply agree with his opinion, so you start testing. The null hypothesis is: The population mean grade is 70%. This is a hypothesized value. The alternative hypothesis is: The population mean grade is not 70%. You can see how both of them are denoted, below.That is the true population mean.The idea is the following. We are standardizing or scaling the sample mean we got. If the sample mean is close enough to the hypothesized mean, then Z will be close to 0. Otherwise, it will be far away from it. Naturally, if the sample mean is exactly equal to the hypothesized mean, Z will be 0.In all these cases, we would accept the null hypothesis.When we calculate Z, we will get a value. If this value falls into the middle part, then we cannot reject the null. If it falls outside, in the shaded region, then we reject the null hypothesis. That is why the shaded part is called: rejection region, as you can see below.Now these are values we can check from the z-table. When α is 0.025, Z is 1.96. So, 1.96 on the right side and minus 1.96 on the left side. Therefore, if the value we get for Z from the test is lower than minus 1.96, or higher than 1.96, we will reject the null hypothesis. Otherwise, we will accept it.That’s more or less how hypothesis testing works. We scale the sample mean with respect to the hypothesized value. If Z is close to 0, then we cannot reject the null. If it is far away from 0, then we reject the null hypothesis.In this situation, the rejection region is on the right side. So, if the test statistic is bigger than the cut-off z-score, we would reject the null, otherwise, we wouldn’t.