The inductive inference consists in arriving at a decision to accept or reject a null hypothesis (H0) after inspecting only a sample from it. As such, an element of risk - the risk of taking wrong decisions is involved. In any test procedure the four possible mutually disjoint and exhaustive decisions are:
- Reject Null Hypothesis when actually it is not true, i.e. when Null is false
- Accept Null when it is true
- Reject Null when it is true
- Accept Null when it is false
The decision in (I) & (ii) are correct decisions while the decisions (iii) & (iv) are wrong decisions.
These decisions may be expressed in the following dichotomous table:
Thus, in testing the hypothesis we are likely to commit two types of errors. The error of rejecting Null when Null is true is known as a Type I error and the error of accepting Null when Null is false (i.e. Alternate is true) is known as Type II error
Remark: Type I & Type II error:
- We make type I error by rejecting a true null hypothesis
- We make type II error by accepting a wrong null hypothesis
If we make
P (Reject Null when it is true) = P (Type I Error) = Alpha (a)
P (Accept Null when it is wrong) = P (Type II Error) = Beta (b.)
Then a & b are also called the sizes of Type I error & Type II error respectively
In the terminology of Industrial Quality Control while inspecting the quality of a manufactured lot, the Type I error amounts to rejecting a good lot and Type II error amounts to accepting a bad lot.
Accordingly,
a = P (Rejecting a good lot)
b= P (Accepting a bad lot)
The sizes of Type I & Type II errors are also known as Producer's risk & Consumer's risk respectively.
An ideal test procedure would be one which is so planned as to safeguard against both these errors. But, practically, in any given problem, it is not possible to minimize both these errors simultaneously. An attempt to decrease a results in an increase in b & vice versa. In practice, in most of decision-making problems in business and social sciences, it is more risky to accept a wrong hypothesis than to reject a correct one, i.e., consequences of type II error are likely to be more serious than the consequences of type I error. Since for a given sample, both the errors cannot be reduced simultaneously, a compromise is made by minimizing more serious errors after fixing up the less serious error.
Thus, we fix a, the size of type I error and then try to obtain a criterion which minimizes b, the size of type II error.
Obviously, when Null is true, it ought to be accepted. Hence, minimizing b amounts to maximizing (1-b.), this is called the power of the test. Hence, the usual practice in testing of hypothesis is to fix a the size of type I error and then try to obtain a criterion which minimizes b, the size of type II error or maximizes (1-b.), the power of the test.
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