# 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:

1. An intelligent student is not promoted to the next class.
2. A good player is not allowed to play in the match.
3. An innocent person is punished.
4. A faultless driver is punished.
5. 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.