testing of hypothesis-1

8/6/2019 Testing of Hypothesis-1

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Testing of Hypothesis

Business Mathematics and Statistics MBA

(FT) I


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Introduction

Inferential statistics is concerned with estimatingthe true value of population parameters usingsample statisticsThere are three techniques of inferential statistics-

Point estimationConfidence interval the interval which is likely to

contain the true parameter valueDegree of confidence associated with a parameter valuewhich lies within an interval.


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Introduction

This information helps decision-maker in determining aninterval estimate of a population parameter value with thehelp of sample statistic along with certain level of confidence of the interval containing the parameter value.Such an estimate is helpful for drawing statisticalinference about the characteristic of the population if interestAnother way of estimating the true value of population

parameters is to test the validity of the claim (assertion or statement) made about this true value using samplestatistics.


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Hypothesis

A statistical hypothesis isan assumption about any aspect of a populationis simply a quantitative statement about population

a claim (assertion, statement, belief or assumption) about anunknown population parameter value

It could be the parameters of a distribution like mean of a Normal distribution, describing the population, the

parameters of two or more populations, correlation or association between two or more characteristics of a population like age and height, etc.


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Hypothesis: Example

1. A judge assumes thata person charged with a crime is innocent

and subject this assumption (hypothesis) to averification by reviewing the evidence and hearingtestimony before reaching to a verdict

2. A pharmaceutical company claims thatThe efficacy of a medicine against a disease that 95 percent of

all persons suffering from the said disease get cured3. An investment company claims that the average return

across all its investments is 20 percent


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Hypothesis Testing

To test such claims or assertions statistically, sampledata are collected and analyzedOn the basis of sample findings the hypothesized value

of the population parameter is either accepted or rejected.The process that enables a decision maker to test thevalidity (or significance) of this claim by analysing the

difference between the value of sample statistic and thecorresponding hypothesized population parameter value, is called hypothesis testing


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Procedure for Hypothesis Testing

1. S tate the N ull Hypothesis (H 0) and Alternative Hypothesis (H 1)

2.State the Level of Significance,

3. Establish Critical or Rejection Region4. Select the Suitable Test of Significance or Test

Statistic5. Formulate a Decision Rule to Accept Null

Hypothesis


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Null Hypothesis

A definite statement about the population parameter(s)Such a statistical hypothesis which is under test, isusually a hypothesis of no difference and hence is called

null hypothesisA hypothesis which is the hypothesis of no difference isnull hypothesisThe null hypothesis presumes that there is no difference

between sample statistic and the parameter valueExample: H 0: = 0.


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Alternative Hypothesis

Any hypothesis which is complementary to the nullhypothesis is called an alternative hypothesis . I t isusually denoted by H 1

Acceptance or rejection of null hypothesis is meaningfulonly when it is being tested against a rival hypothesiswhich should rather be explicitly mentionedExample:

H1: 0.H1: > 0.H1: < 0.


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Level of Significance ( )

It is specified before the samples are drawn, sothat the results obtained should not influence thechoice of the decision-maker It is specified in terms of the probability of nullhypothesis H 0 being wrong


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Level of Significance ( )

The level of significance defines the likelihoodof rejecting a null hypothesis when it is true, i.e.it is the risk a decision-maker takes of rejecting

the null hypothesis when it is really trueThe guide provided by the statistical theory isthat this probability must be small.Traditionally

= 0.05 is selected for consumer research projects = 0.01 for quality assurance and = 0.10 for political polling


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Critical Region or Rejection Region

The area under the sampling distribution curve of thetest statistic is divided into two mutually exclusiveregions (areas). These regions are called the acceptance

region and the rejection (critical) region.The acceptance region is a range of values of thesample statistic spread around the null hypothesized

population parameter.

If values of the sample statistic fall within the limits of acceptance region, the null hypothesis is accepted,otherwise it is rejected.


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The rejection region is the range of samplestatistic values within which if values of thesample statistic falls (i.e. outside the limits of the acceptance region), then null hypothesis isrejected.The value of the sample statistic that separates

the regions of acceptance and rejection is calledcritical value .


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The size of the rejection region is directly related to thelevel of precision to make decisions about the

population parameter.

Decision rule concerning null hypothesis are as follows:If probability (H 0 is true) , then reject H 0.If probability (H 0 is true) > , then accept H 0.

If probability of H 0 being true is less than or equal to thesignificance level, then reject H 0, otherwise accept H 0.The level of significance is used as the cut-off pointwhich separates the area of acceptance from the are of rejection


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Test Statistic

The tests of significance or test statistics are classifiedinto two categoriesParametric testsNon-parametric tests

Parametric testsare more powerful because their data are derived from intervaland ratio measurementsare the tests of choice provided certain assumptions are metAssumption for these tests are as follows:

The selection of any element from the population should not affectthe chance for any other to be included in the sample to be drawnfrom the populationThe samples should be drawn from normally distributed populationsPopulations under study should have equal variances


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Test Statistic

Non-parametric testsare used to test hypotheses with normal and ordinaldatahave few assumptions and do not specify normallydistributed populations or homogeneity of variance


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Selection of Test Statistic

For choosing a particular test statistic following threefactors are considered:

Whether the test involves one sample, two samples, or k

samples?Whether two or more samples used are independent or related?Is the measurement scale nominal, ordinal, interval, or ratio?

Further, it is also important to know:

Sample sizeThe number of samples and their sizeWhether data have been weighted


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Formulate a Decision Rule to Accept

Null HypothesisCompare the calculated value of the test statisticwith the critical value (also called standard tablevalue of the test statistic).The decision rules for null hypothesis are asfollows:

Accept H 0 if the test statistic value falls within the

area of acceptance, i.e. if calculated absolute value of a test statistic is less than or equal to its critical valueReject otherwise

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