USA: +1-585-535-1023

UK: +44-208-133-5697

AUS: +61-280-07-5697

Statistical Hypothesis and Related Terms

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 H0. For example,

 

H0 : µ = µ0, or, H0 : µ1 - µ2 = K, H0 : σ2 = σ0 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 > µ0, or, H1 : µ < µ0    or    H1 : µ≠ µ0 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                     H1 : µ > µ0 then it is called right tailed test.

If                     H1: µ < µ0,  then it is called left tailed test.

If                     H1: µ ≠ µ0, 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 = µ0 , σ = σ 0

Otherwise it is called composite hypothesis. e.g., µ < µ0 , σ > σ 0 etc.

Steps in testing hypothesis:

  1. Set up H0.
  2. Set up H1.
  3. Set up test statistic.
  4. Set up the level of significance and critical value(s) using statistical table.
  5. Compute the value of statistic using sample drawn from population.
  6. Take decision. If the calculated value of test statistic lies in the critical region, reject H0 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 H0 : 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 H0 : P = 0.1.

Accept H0 : If the no. of defectives found in the sample is less than equal to 6.

Reject H0 : 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) p7Q +(8, 8 ) QQ

=          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