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 acceptance region or rejection region. If the calculated value of 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 two errors which are called Type I and Type Il-errors.
The null hypothesis may be true but it may be rejected. This is an error and is called 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 Type 1 error. The probability is that is accepted when is true. It is called correct decision. We can say that 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 the match.
- an innocent person is punished.
- a driver is punished for no fault of him.
- a good worker is not paid his salary in time.
These are the examples from practical life. These examples are quoted to make a point clear to the students.
The probability of making 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 that 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 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 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 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 away from the sampling distribution under . There will be hardly any test-statistic 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 Type II error is denoted by . 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.