Critical Region for z-test

A calculation of a critical region (or a rejection region) is one of the steps of a testing of a statistical hypothesis. When we have a value of a test statistics and a critical region, we can decide about rejecting or non-rejecting of a null hypothesis. There is a rule which we can call a “golden rule of the hypothesis testing.

If a value of a statistics is an element of a critical region, the null hypothesis is rejected.

And consequently, if a value of a statistics is not an element of a critical region, the null hypothesis is not rejected.

We will show the computation of the critical region on z-test, which is described in this article. Just to give you a quick review: The null hypothesis of the z-test is that a expected value of a data equals to a given number (in our case 190). The alternative hypothesis is that the expected value differs from the given number (does not equal 190). We assume two-tailed test, one-tailed text will be described in another article.

As is written in the article about z-test, each test has its statistics and each statistics has a statistical distribution. The statistics of z-test has a normalized normal (Gauss) distribution. The distribution function of the normal distribution is defined for all real numbers, but only a part of them make up the critical region. Our task is to identify this part.

There is a simple logic behind this. Let’s look once more on the formula of statistics:

$Z = \frac{ \bar{x} - \mu_0}{\sigma} \cdot \sqrt{n} \, ,$

Now let’s assume that null hypothesis is true, i.e. the expected value of the data equals 190 ($\mu_0 = 190$). It is highly probable that the average value $\bar{x}$ of a random sample is close to 190. The statistics has a difference $\bar{x} - \mu_0$ in the numerator so if $\bar(x)$ is close to $\mu_0$ then the value of statistics is close to zero.

On the other hand, the average value of a random sample might be significantly different from 190, but it is rather unlikely. If there is a big difference between the hypothetical expected value and the average value of the random sample, then the value of the statistics is far from zero and it can be both positive or negative. The further is the value of the statistics from zero the less probable it is.

So we will cut of the least probable values of the statistics and we will add them to the critical region. But how much of the values? It depends on a level of significance (denoted as $\alpha$). The level of significance basically says what percentage of the least probable values we will add to the critical region. Usually, we add there 1 %, 5 % or 10 %.

We can depict that using a probability density function (PDF). On a figure below, you can see PDF of the normalized normal distribution with critical regions for three values of $\alpha$. As you can see, the critical region consists of the most extreme values. The higher is $\alpha$ the larger get the critical region. Critical region for each $\alpha$ consists of two equally sized parts – one on the left and one on the right.

So the formula for the critical region is:

$W = ( - \infty, u_{\frac{\alpha}{2}} \rangle \cup \langle u_{1 - \frac{\alpha}{2}}, \infty ) \, .$

If we substitute to the formula, we see that for $\alpha = 1 %$ the

$W = ( - \infty, -2.57583 \rangle \cup \langle 2.57583, \infty ) \, ,$

for $\alpha = 5 %$

$W = ( - \infty, -1.95996 \rangle \cup \langle 1.95996, \infty ) \, ,$

and finally for $\alpha = 10 %$

$W = ( - \infty, -1.64485 \rangle \cup \langle 1.64485, \infty ) \, .$