Class notes Statistics

STATISTICS




STATISTICS

TESTING OF HYPOTHESIS

Population:
The totality of any finite (or) infinite collection of individuals with which we are concerned,
possessing a variable character is called a Population.

Sample:
A finite subset of a population is called a sample and the process of selection of samples is called
sampling.

Large and Small samples:
The number of elements in a sample is greater than or equal to 30, then the sample is called a
large sample and if it is less than 30, then the sample is called a small sample.

Parameters:
Statistical constants like mean  , variance  2 … computed from a population are called
parameters of the population.

Statistics:
Statistical constants like mean x , variance s 2 … computed from a sample are called sample
statistics (or) statistics.

Sampling Distribution:
The probability distribution of a statistic is called the sampling distribution.
Example, If we take ' k ' samples each of size n1, n2, n3, ...nk , we can find their means x1 , x2 , x3 ...xk .
This set of sample means is called a sampling distribution of the sample statistic x .

Estimation:
Some characteristic (or) feature of the population in which we are interested may be completely
unknown to us and we may like to make a guess about this characteristic entirely on the basis of a
random sample drawn from the population. This type of problem is known as the problem of
estimation.

Testing of Hypothesis:
Some information regarding the characteristic or feature of the population may be available to us
and we may like to know whether the information can be accepted in the light of the random
sample drawn from the population and if it can be accepted, with what degree of confidence it can
be accepted. This type of problem is known as the problem of testing of hypothesis.

Standard Error (S.E):
The standard deviation of the sampling distribution of a statistic is known as its standard error and
it is denoted by S.E.


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Tests of Significance:
A very important aspects of the sampling theory is the study of the tests of significance which
enable us to decide on the basis of the sample results if
(i) The deviation between the observed sample statistic and the hypothetical parameter value is
significant.
(ii) The deviation between two sample statistics is significant.

Null Hypothesis (H 0 ):
For applying the test of significance, we first set up of a hypothesis - a definite statement about
the population parameter. Such a hypothesis is usually a hypothesis of no-difference and it is
denoted by H 0 .

Alternative Hypothesis (H 1 ):
Any hypothesis which is complementary to the null hypothesis is called an alternative hypothesis
and is denoted by H 1 .

Critical Region:
For a test statistic, the area under the probability curve which is normal is divided into two
regions namely the region of acceptance of H 0 and the region of rejection of H 0 . The region in
which H 0 is rejected is called the critical region. The area of the critical region is  , the level of
significance. The region in which H 0 is accepted is called the acceptance region.

Critical ratio:
t  E (t )
For the statistic E, we find the relation Z  which is called the critical ratio. Here Z
s.E (t )
follows a normal distribution with mean zero and S.D. unity.

Errors in sampling:
After applying a test of significance a decision is to be taken to accept (or) reject the null
hypothesis H 0 . There is always same possibility of committing error in taking a decision. These
errors are of two types: (i) Type I error (ii) Type II error
(i) Type I Error: The rejection of the null hypothesis H 0 when it is really true is called
Type I Error. It is also known as producer’s risk.
(ii) Type II Error: The acceptance of the null hypothesis H 0 when it is false is called
Type II Error. It is also known as consumer’s risk.

Level of Significance:
The probability of Type I error is called the level of significance of the test and is denoted by  .
We usually take either   5 % (or) 1 %

One tailed test:
In a test of any statistical hypothesis, if the alternative hypothesis is one sided then it is called one
tailed test. It may be right tailed (or) left tailed test. Example. H 0 :   0 .


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Two tailed test:
In a test of statistical hypothesis, if the alternative hypothesis is two sided, then it is called two
tailed test. Ex. H 0   0 ,. H1 :   0

Note:
1. 5% level of significance means we are 95% confident that we have made the right
decision.
2. 1% level of significance mean we are 99% confident that we have made the correct
decision.
3. The critical values for some standard LOS’s are given in the following table both for two-
tailed and one tailed tests;
Nature of 10% 5% 2% 1%
Test 0.1 0.05 0.02 0.01
Two tailed z =1.645 z =1.96 z =2.33 z =2.58

Right tailed
1.28 1.645 2.055 2.33
Left tailed
-1.28 -1645 -2.055 -2.33

Procedure for Solving Testing of Hypothesis:
1. State the null hypothesis H 0 .
2. Decide the alternate hypothesis H1 .
3. Choose the level of significance(LOS) ‘  ’ 5% or 1% 
t  E (t )
4. Compute the test statistic Z  .
s.E (t )
5. Compare the computed value of Z with the table value of and decide the acceptance (or)
the rejection of H 0 . If Z  Z , H 0 is accepted (or) H1 is rejected.
Some Notations:
N  Population Size n  Sample Size
  Population Mean x  Sample Mean
 2  Population Variance s 2  Sample Variance
P  Population Proportion p  Sample Proportion
  Population S.D s  Sample S.D

Confidence interval (or) Confidence (Fudicial) Limits:
The interval within which the parameter is expected to lie is called the confidence interval. The
end points of the interval are called confidence limits.




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