# Type-I Error and Type-II Error

The null hypothesis $${H_o}$$ is accepted or rejected on the basis of the value of the test-statistic, which is a function of the sample. The test statistic may land in the acceptance region or rejection region. If the calculated value of the test-statistic, say $$Z$$, is small (insignificant) i.e., $$Z$$ is close to zero or we can say $$Z$$ lies between $$ – {Z_{\alpha /2}}$$ and $${Z_{\alpha /2}}$$ is a two-sided alternative test $$\left( {{H_1}:\theta \ne {\theta _o}} \right)$$, the hypothesis is accepted. If the calculated value of the test-statistic $$Z$$ is large (significant), $${H_o}$$ is rejected and $${H_1}$$ is accepted. In this rejection plan or acceptance plan, there is the possibility of making any one of the following two errors, which are called Type-I and Type-II errors.

__Type-I Error__

The null hypothesis $${H_o}$$ may be true but it may be rejected. This is an error and is called a Type-I error. When $${H_o}$$ is true, the test-statistic, say $$Z$$, can take any value between $$ – \infty $$to$$ + \infty $$ . But we reject $${H_o}$$ when $$Z$$ lies in the rejection region while the rejection region is also included in the interval $$ – \infty $$to$$ + \infty $$. In a two-sided $${H_1}$$ (like$$\theta \ne {\theta _o}$$), the hypothesis is rejected when $$Z$$ is less than $$ – {Z_{\alpha /2}}$$ or $$Z$$ is greater than $${Z_{\alpha /2}}$$. When $${H_o}$$ is true, $$Z$$ can fall in the rejection region with a probability equal to the rejection region $$\alpha $$. Thus it is possible that $${H_o}$$ is rejected while $${H_o}$$ is true. This is called a Type-I error. The probability is $$\left( {1 – \alpha } \right)$$ that $${H_o}$$ is accepted when $${H_o}$$ is true. This is called a correct decision. We can say that a Type-I error has been committed when:

- An intelligent student is not promoted to the next class.
- A good player is not allowed to play in the match.
- An innocent person is punished.
- A faultless driver is punished.
- A good worker is not paid his salary in time.

These are some examples from practical life. These examples are quoted to provide clarity.

**Alpha $$\alpha $$**

The probability of making a Type-I error is denoted by $$\alpha $$(alpha). When a null hypothesis is rejected, we may be wrong in rejecting it or we may be right in rejecting it. We do not know if $${H_o}$$ is true or false. Whatever our decision will be, it will have the support of probability. A true hypothesis has some probability of rejection and this probability is denoted by $$\alpha $$. This probability is also called the size of the Type-I error and is denoted by $$\alpha $$.

__Type-II Error__

The null hypothesis $${H_o}$$ may be false but it may be accepted. It is an error and is called a Type-II error. The value of the test-statistic may fall in the acceptance region when $${H_o}$$ is in fact false. Suppose the hypothesis being tested is $${H_o}:\theta = {\theta _o}$$ and $${H_o}$$ is false and the true value of $$\theta $$ is $${\theta _1}$$ or$${\theta _{{\text{true}}}}$$. If the difference between $${\theta _1}$$ is very large then the chance is very small that $${\theta _o}$$(wrong) will be accepted. In this case the true sampling distribution of the statistic will be quite far away from the sampling distribution under $${H_o}$$. There will be very few test-statistics which will fall in the acceptance region of $${H_o}$$. When the true distribution of the test-statistic overlaps the acceptance region of $${H_o}$$, then $${H_o}$$ is accepted though $${H_o}$$ is false. If the difference between $${\theta _o}$$ and $${\theta _1}$$ is small, then there is a high chance of accepting $${H_o}$$. This action will be an error of Type-II.

**Beta$$\beta $$**

The probability of making a Type-II error is denoted by $$\beta $$. A Type-II error is committed when $${H_o}$$ is accepted while $${H_1}$$ is true. The value of $$\beta $$ can be calculated only when we happen to know the true value of the population parameter being tested.