The null hypothesis 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 , is small (insignificant) i.e., is close to zero or we can say lies between and is a two-sided alternative test , the hypothesis is accepted. If the calculated value of the test-statistic is large (significant), is rejected and 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.
The null hypothesis may be true but it may be rejected. This is an error and is called a Type-I error. When is true, the test-statistic, say , can take any value between to . But we reject when lies in the rejection region while the rejection region is also included in the interval to. In a two-sided (like), the hypothesis is rejected when is less than or is greater than . When is true, can fall in the rejection region with a probability equal to the rejection region . Thus it is possible that is rejected while is true. This is called a Type-I error. The probability is that is accepted when 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.
The probability of making a Type-I error is denoted by (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 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 . This probability is also called the size of the Type-I error and is denoted by .
The null hypothesis 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 is in fact false. Suppose the hypothesis being tested is and is false and the true value of is or. If the difference between is very large then the chance is very small that (wrong) will be accepted. In this case the true sampling distribution of the statistic will be quite far away from the sampling distribution under . There will be very few test-statistics which will fall in the acceptance region of . When the true distribution of the test-statistic overlaps the acceptance region of , then is accepted though is false. If the difference between and is small, then there is a high chance of accepting . This action will be an error of Type-II.
The probability of making a Type-II error is denoted by . A Type-II error is committed when is accepted while is true. The value of can be calculated only when we happen to know the true value of the population parameter being tested.