In a testing problem, the null hypothesis is not rejected in favor of the alternate hypothesis if the calculated value of the test statistic (denoted by Tcalc, say) chosen falls within the region of acceptance, denoted by W.

If the value of Tcalc falls outside W, then the null hypothesis is rejected in favor of the alternate hypothesis.

**Type I Error**

Such a case may arise wherein Tcalc W, still the null hypothesis gets rejected.

This type of error is known as type I error.

**Definition:**

The error committed by rejecting a true null hypothesis is known as a type I error.

**Type II Error**

It may also happen that Tcalc W, but still, the null hypothesis does not get rejected in favor of the alternate hypothesis. This type of error is known as type II error.

**Definition:**

The error committed by accepting a false null hypothesis is known as a type II error.

Situation
Decision | H0 True | H0 False |

H0 Rejected | Type I Error | Correct Decision |

H0 Not Rejected | Correct Decision | Type II Error |

In a testing problem, the choice of the null hypothesis depends highly should be made keeping in mind both types of errors. A test is termed as good if both types of errors are kept under control since, for practical purposes, it is impossible to get rid of any errors.

Now, it is assumed that the commission of the errors is a random event. As such, the experimenters can easily calculate the probabilities associated with them.

Since the problem of hypothesis testing consists of a missing parameter (say ), the probabilities will also depend on it.

**Probability Associated With Type I Error**

The probability of type I error associated with is given by:

P [Type I Error] =P [(X1,X2, X3, ..., XN) W]= P(W), 0

Where

X1,X2, X3, ..., XN denotes the population under study

W denotes the acceptance region

0 denotes a specified proper subset of the parameter space

Let be any number such that 0<<1. This value indicates the level at which the probability of type I error should be kept for a good test. So we have,

P(W) = , 0 is known as a test's significance level.

**Probability Associated With Type II Error**

The probability of type Ii error associated with is given by:

P [Type II Error] =P [(X1,X2, X3, ..., XN) A]= P(A), -0

Where

X1,X2, X3, ..., XN denotes the population under study

A denotes the rejection region

-0 denotes a specified proper subset of the parameter space

**Relationship Amid the Probabilities of Type I and Type II Error**

The region of acceptance, W, and the rejection region A can be thought of as two sets in the cartesian plane. The culmination of these two sets forms the entire range of values for the test.

Both these regions are compliments of each other, i.e., W=AC

Where Ac is the set complimentary to A.

So, the probability of type II error can also be written as:

P(A) = P(WC)= 1-P(W)

For -0

The probability () =P(W) is a function of () is called the power function of the test.

We have:

() = the probability of type I error associated with , 0

() = 1 - the probability of type II error associated with , -0

The power function is used to judge the nature of the whole test.