tests of hypothesis [compatibility mode].pdf

8/14/2019 Tests of Hypothesis [Compatibility Mode].pdf

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Tests of Hypothesis


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Sampling Theory Population: A statistical population is the

set of all possible measurements on datacorresponding to the entire collection ofunits for which an inference is to be

made. Another definition: Population is an

aggregate of objects, animate or

inanimate, under study.

Population may be finite or infinite.

Complete enumeration of population inim racticable.


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Sample

A sample is a finite subset ofpopulation which is collected to drawan inference about the population.

r usamples is called sampling.


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Parameter and Statistic

Statistical measures calculated on thebasis of population are calledparameters.

on the basis of sample observationsare called statistics.

Example: Mean of a population is a

parameter and mean of a sample is astatistic.


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Sampling distribution of thestatistic

The probability distribution of somestatistic of a random sample is calledthe sampling distribution of the

.


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Standard Error (S.E)

The standard deviation of a statistic isknown as its standard error.

Utility of S.E: It forms the basis of the

y . If t is any statistic, then for large

samples z=[t-E(t)]/S.E which follows

std. Normal distribution.


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Some information regarding thecharacteristic of the population maybe known to us and we may like towhether the information can be

accepted w.r.to random sampledrawn from the population and if itcan be accepted, to what degree of

confidence it can be accepted. Thistype of problem is known as theproblem of testing of hypotheses.


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Statistical Hypotheses

Assumptions which are made aboutthe nature of the population or aboutthe value of some parameter of the

true) on the basis of a randomsample are called statisticalhypotheses.


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Types of Hypotheses

Null hypothesis: Hypothesis of nodifference is called a null hypothesis ,that is, a hypothesis which assumes

between sample statistic andpopulation parameter or no differencebetween two sample statistics iscalled the null hypothesis. We denotethe null hypothesis by H0.


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Alternative hypothesis: A hypothesiswhich is complementary to the nullhypothesis is called an alternative

1.


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Test of hypothesis

A procedure for deciding whether toaccept or reject the null hypothesis iscalled the test of hypothesis.


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Test of significance

If 0

is a parameter of the populationand is the corresponding samplestatistic, then usually there will be

0this difference is different for differentsamples, since is based on sampleobservations. Such a difference whichis due to sampling fluctuations iscalled insignificant difference.


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The difference that arises due to the reason

that either the sampling procedure is notpurely random or that the sample has notbeen drawn from the given population is

known as significant difference. The procedure of testing whether the

difference between 0 and is significant or

not is called the test of significance.


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Critical region and level ofsignificance

A region corresponding to a statistic tin the sample space S which amountsto rejection of null hypothesis is

region of rejection.

The region complementary to the

region of rejection is called theregion of acceptance.


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The probability that a random valueof the statistic t belongs to the criticalregion is known as the level of

.expressed as a percentage.


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The levels of significance usuallyemployed in testing of hypothesesare 1% and 5%.

w y x vbefore collecting the sampleinformation.

In any test, the critical region isrepresented by a portion of the areaunder the probability curve of the


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Sampling distribution of the teststatistic.


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Errors in Hypotheses testing

Due to the wrong judgement of LOS,the region of rejection becomes largerand the probability of rejecting a null

greater. In this case we arrive at thefollowing two types of errors:

Type-I error:Rejecting H0 when it istrue.

This is similar to a good product


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Good product being rejected by aconsumer and so Type-I error is alsoknown as producers risk.

Type-II error: Accepted H0 when it isfalse.

This error is similar to that of

accepted a product of inferior qualityand hence it is also known asconsumers risk.

It may be noted that the probabilityof committing Type-I error is theLOS.


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One-tailed and Two-tailed tests

If 0 is a population parameter and is the corresponding sample statisticand if we set up a null hypothesis H0:

0 = , r vhypothesis H1 will be any one of thefollowing:

(i) H1: 0 ;(ii)H1: > 0 ;(iii) H1: < 0.


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When H0 is tested while H1 is two-tailed alternative, the test ofhypothesis is called a two-tailed test.

: -and two-tailed tests depend upon thenature of the alternative hypothesis

and the choice of the appropriate H1depends on the situation and thenature of the problem concerned.


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Critical values or significantvalues

The value of test statistic z which

separates the critical region andacceptance region is called the critical

is denoted by z. It depends up on(i)

the LOS used and (ii) the alternativehypothesis (one tailed or two tailed).


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tests of hypothesis [compatibility mode].pdf

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