There are many problems in which we have to make decisions about a statistical population on the basis of sample observations. To reach a decision, we make an assumption or statement about the population which is known as a *statistical hypothesis.*

For example, *(i) *the average marks of students in a university is 77% (ii) the average lifetime of a certain tires is at least 25,000 miles, *(iii) *difference of resistance between two types of electric wires is 0.025 ohm, etc.

The hypothesis which we are going to test for possible rejection under the assumption is called *'Null Hypothesis' *which is usually denoted by H_{0}. For example,

H_{0} : µ = µ_{0}, or, H_{0} : µ_{1} - µ_{2} = K, H_{0} : σ^{2} = σ_{0} etc.

Any hypothesis which is taken as complementary to the null hypothesis is called an *'Alternative *- *Hypothesis' *and is usually denoted by H1. For example,

H1 : µ > µ_{0}, or, H1 : µ < µ_{0} or H1 : µ≠ µ_{0} etc.

The next step is to set up a statistic. While accepting or rejecting a hypothesis we commit two types of errors :

**Type I Error :** Reject H0 when it is true.

**Type II Error :** Accept H0 when it is false.

If we consider

P [ type I error] = α,

P [ type II error ] = β

then α and β are referred to as producer's risk and consumer's risk respectively.

**Critical region.** A region corresponding to a statistic which amounts to rejection of H0 is

termed as critical region or region of rejection.

** **

**Level of significance (α).** This is a probability that a random value of the statistic belongs to the critical region. In other words, it is the size of the critical region. Usually, the level of significance is taken as 5 % or 1%. So a = P [Type I error].

**Critical value.** The value which separates the critical region and the acceptance region is called critical value which is set by seeing the alternative hypothesis.

**Type of tests**. It is determined based on the alternative hypothesis. For example,

If H_{1 }: µ > µ_{0} then it is called right tailed test.

If H1: µ < µ_{0, } then it is called left tailed test.

If H1: µ ≠ µ_{0,} then it is called both tailed test (/two tailed test).

**Simple and composite hypothesis.** If all the parameters are completely specified, the hypothesis is called simple, *e.g., *µ** **= µ_{0 , }σ = σ _{0}

Otherwise it is called composite hypothesis. *e.g., *µ** **< µ_{0 , }σ > σ _{0 }etc.

Steps in testing hypothesis:

- Set up H0.
- Set up H1.
- Set up test statistic.
- Set up the level of significance and critical value(s) using statistical table.
- Compute the value of statistic using sample drawn from population.
- Take decision. If the calculated value of test statistic lies in the critical region, reject H
_{0 }*i.e.,*the assumption under the null hypothesis cannot be accepted. If the calculated value of test statistic lies in the accepted region*i.e.,*outside the critical region, accept H0 ,*i.e.,*the assumption under H0 can be taken as true value of the parameter.

**Power function of the test.** Let β = P [ Type II error ]. Then 1 - 13 is called the power function of testing H0 against H1.

**Example 1.** *The fraction of defective items in a large lot is P To test H _{0} : P *=

*0.1, one considers the number of defectives in a sample of 8 items and accept H0 if the number of defectives is less than 6. Otherwise he rejects the hypothesis. What is the probability of type I error of this test ? What is the probability of type II error ?*

** **

**Solution.** Given H_{0} : P = 0.1.

Accept H_{0} : If the no. of defectives found in the sample is less than equal to 6.

Reject H_{0 }: If the no. of defectives found in the sample is 7 or 8.

Here, the no. of defectives in a lot follows binomial distribution with *n *= 8 and P = 0.1,

Q = 1 - P = 0.9.

P [Type I error] = Probability of rejecting H0 when H0 is true

= Probability of 7 or 8 defectives

= (8, 7) p^{7}Q +(8, 8 ) Q^{Q}

= 8 (0.1)^{7} (0.9) + (0.1)^{8}

= 0.00000073.

P [Type II error] = Probability of accepting H0 when H0 is false

= Probability of defectives which are less than equal to 6

= 1 - [ Probability of 7 or 8 defectives]

= 1 - 0.00000073

= 0.99999927