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# Basis of statistical inference

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### Basis of statistical inference

1. 1. Basis of statistical inferenceBasis of statistical inference Statistical inference is the branch of statisticsStatistical inference is the branch of statistics which is concerned with using probability conceptwhich is concerned with using probability concept to deal with uncertainly in decision makingto deal with uncertainly in decision making.. It refers to the process of selecting and using aIt refers to the process of selecting and using a sample to draw inference about population fromsample to draw inference about population from which sample is drawnwhich sample is drawn..
2. 2. Statistical Inference Estimation of population value Testing of hypothesis Point estimation Range estimation Mean, proportion estimation Confidence interval estimation
3. 3. Hypothesis and hypothesis testingHypothesis and hypothesis testing • During investigation there is assumption andDuring investigation there is assumption and presumption which subsequently in study must bepresumption which subsequently in study must be proved or disproved.proved or disproved. • Hypothesis is a supposition made from observation.Hypothesis is a supposition made from observation. On the basis of hypothesis we collect the data.On the basis of hypothesis we collect the data. • Hypothesis is a tentative justification, the validity ofHypothesis is a tentative justification, the validity of which remains to be tested.which remains to be tested. Two hypothesis are made to draw inference fromTwo hypothesis are made to draw inference from Sample valueSample value-- A.A. Null hypothesis or hypothesis of no difference.Null hypothesis or hypothesis of no difference. B.B. Alternative hypothesis of significant difference.Alternative hypothesis of significant difference.
4. 4. Hypothesis and hypothesis testingHypothesis and hypothesis testing The null hypothesis is symbolized as Ho andThe null hypothesis is symbolized as Ho and alternative hypothesis is symbolized as Halternative hypothesis is symbolized as H11 or Hor HAA.. In hypothesis testing we proceed on the basis ofIn hypothesis testing we proceed on the basis of null hypothesis. We always keep alternativenull hypothesis. We always keep alternative hypothesis in mindhypothesis in mind.. The null hypothesis and the alternativeThe null hypothesis and the alternative hypothesis are chosen before the sample ishypothesis are chosen before the sample is drawndrawn..
5. 5. Null hypothesisNull hypothesis A null hypothesis or hypothesis of no differenceA null hypothesis or hypothesis of no difference (Ho) between statistic of a sample and(Ho) between statistic of a sample and parameter of population or between statistic ofparameter of population or between statistic of two samples nullifies the claim that thetwo samples nullifies the claim that the experimental result is different from or betterexperimental result is different from or better than the one observed already. In other wordsthan the one observed already. In other words null hypothesis states that the observednull hypothesis states that the observed difference is entirely due to sampling error thatdifference is entirely due to sampling error that is it occurs purely by chanceis it occurs purely by chance..
6. 6. Examples of null form hypothesisExamples of null form hypothesis  There is no difference between the operationalThere is no difference between the operational procedures of open prostatectomy and TURP.procedures of open prostatectomy and TURP.  There is no difference between open operation andThere is no difference between open operation and transsphenoidal approach.transsphenoidal approach.  Para median incision is as good as pfannenstiel’sPara median incision is as good as pfannenstiel’s incision in lower segment Caesarian section.incision in lower segment Caesarian section.  There is no difference in the incidence of measlesThere is no difference in the incidence of measles between vaccination and non-vaccination children.between vaccination and non-vaccination children.  Drugs chlorampenicol is as good as drugDrugs chlorampenicol is as good as drug cotrimoxazol in treating enteric fever.cotrimoxazol in treating enteric fever.  Drug ‘Z’ is not effective in curing malaria.Drug ‘Z’ is not effective in curing malaria.
7. 7. The alternative hypothesisThe alternative hypothesis Alternative hypothesis of significant differenceAlternative hypothesis of significant difference states that the sample result is different that isstates that the sample result is different that is greater or smaller than the hypothetical value ofgreater or smaller than the hypothetical value of populationpopulation.. A test of significance such as Z-test, t-test, chi-A test of significance such as Z-test, t-test, chi- square test is performed to accept the nullsquare test is performed to accept the null hypothesis or to reject it and accept thehypothesis or to reject it and accept the alternative hypothesisalternative hypothesis..
8. 8. Characteristics of hypothesisCharacteristics of hypothesis 1.1. Hypothesis should be clear and precise.Hypothesis should be clear and precise. 2.2. Hypothesis should be capable of being tested.Hypothesis should be capable of being tested. 3.3. It should state relationship between variables.It should state relationship between variables. 4.4. It must be specific.It must be specific. 5.5. It should be stated as simple as possible.It should be stated as simple as possible. 6.6. It should be amenable to testing within aIt should be amenable to testing within a reasonable time.reasonable time. 7.7. It should be consistent with known facts.It should be consistent with known facts.
9. 9. Interpreting the result of hypothesisInterpreting the result of hypothesis The hypothesis Ho is true and our test accept itThe hypothesis Ho is true and our test accept it because the result falls within the zonebecause the result falls within the zone acceptance at 5% level of significanceacceptance at 5% level of significance.. The hypothesis Ho is false and test rejects itThe hypothesis Ho is false and test rejects it because the estimate falls in shaded area ofbecause the estimate falls in shaded area of rejectionrejection.. Hypothesis Ho is true still it is rejected, throughHypothesis Ho is true still it is rejected, through the estimate falls in acceptance zone at 5%the estimate falls in acceptance zone at 5% level of significance in plain arealevel of significance in plain area.. The hypothesis Ho is false but it is accepted,The hypothesis Ho is false but it is accepted, through the estimate falls in the zone ofthrough the estimate falls in the zone of rejectionrejection..
10. 10. Zone of acceptance & Zone of rejectionZone of acceptance & Zone of rejection Zone of acceptance- If the results of a sampleZone of acceptance- If the results of a sample falls in the plain area i.e. within the mean+-1.96falls in the plain area i.e. within the mean+-1.96 standard error the null hypothesis is accepted-standard error the null hypothesis is accepted- the area is called zone of acceptancethe area is called zone of acceptance.. Zone of rejection-If the result of a sample falls inZone of rejection-If the result of a sample falls in the shaded area i.e. beyond mean +- 1.96the shaded area i.e. beyond mean +- 1.96 standard error it is significantly different fromstandard error it is significantly different from population value. So null hypothesis is rejectedpopulation value. So null hypothesis is rejected and alternative hypothesis is accepted. Thisand alternative hypothesis is accepted. This area is called zone of rejectionarea is called zone of rejection..
11. 11. P ValueP Value The p value represents the probability that theThe p value represents the probability that the results occurred purely by chance. It is theresults occurred purely by chance. It is the probability of observing the result by chanceprobability of observing the result by chance..  When the p value is between 0.05 and 0.01 theWhen the p value is between 0.05 and 0.01 the result is usually called significant.result is usually called significant.  When p value is less than 0.01, result is oftenWhen p value is less than 0.01, result is often called highly significant.called highly significant.  When p value is less than 0.001 and 0.005,When p value is less than 0.001 and 0.005, result is taken as very highly significant.result is taken as very highly significant.
12. 12. Level of significanceLevel of significance It is the significance probability on the basis of which nullIt is the significance probability on the basis of which null hypothesis is rejected is called the level of significance.hypothesis is rejected is called the level of significance. The limit of the region at which we no longer regard theThe limit of the region at which we no longer regard the chance to be operating is called the level of significance. Itchance to be operating is called the level of significance. It is denoted byis denoted by αα.. If the chance limit is set at mean ±1.96 standard error, itIf the chance limit is set at mean ±1.96 standard error, it means 5% level of significancemeans 5% level of significance.. Level of significance is usually determined in advance beforeLevel of significance is usually determined in advance before testing hypothesistesting hypothesis..  Significance at 5% probability level is indicated as p<0.05.Significance at 5% probability level is indicated as p<0.05.  Significance at 1% probability level is indicated as p<0.01.Significance at 1% probability level is indicated as p<0.01.  Significance at 0.1% probability level is indicated asSignificance at 0.1% probability level is indicated as p<0.001.p<0.001.
13. 13. Type 1 and type 2 errorType 1 and type 2 error When a null hypothesis is tested, there may be fourWhen a null hypothesis is tested, there may be four possible outcomespossible outcomes:: i.i. The null hypothesis is true but our test rejects it.The null hypothesis is true but our test rejects it. ii.ii. The null hypothesis is false but our test accepts it.The null hypothesis is false but our test accepts it. iii.iii. The null hypothesis is true and our test accepts it.The null hypothesis is true and our test accepts it. iv.iv. The null hypothesis is false but our test rejects it.The null hypothesis is false but our test rejects it. Type 1 error –Type 1 error – rejecting null hypothesis when nullrejecting null hypothesis when null hypothesis is true. It is calledhypothesis is true. It is called αα errorerror.. Type 2 error –Type 2 error – accepting null hypothesis when nullaccepting null hypothesis when null hypothesis is false. It is calledhypothesis is false. It is called ββ-error-error..
14. 14. Type 1 error and type 2 error are given inType 1 error and type 2 error are given in tabular formtabular form DecisionDecision Accept hoAccept ho Reject hoReject ho Ho trueHo true CorrectCorrect decisiondecision Type 1 errorType 1 error Ho falseHo false Type 2 errorType 2 error CorrectCorrect decisiondecision
15. 15. Type 1 and type 2 error maybe compared withType 1 and type 2 error maybe compared with social crimesocial crime DecisionDecision Charge karimCharge karim Release karimRelease karim Karim killKarim kill RahimRahim CorrectCorrect decisiondecision ErrorError Karim didn’t killKarim didn’t kill RahimRahim ErrorError CorrectCorrect decisiondecision
16. 16. Type 1 and type 2 error maybe comparedType 1 and type 2 error maybe compared with social crimewith social crime Drugs has noDrugs has no effect accept hoeffect accept ho Drug has noDrug has no effect reject hoeffect reject ho Ho is trueHo is true No error (correctNo error (correct decisiondecision(( Type 1 errorType 1 error Ho falseHo false Type 2 errorType 2 error No error (correctNo error (correct decisiondecision((
17. 17. Type 1 and type 2 error maybe compared withType 1 and type 2 error maybe compared with social crimesocial crime The null hypothesis isThe null hypothesis is TrueTrue FalseFalse Accept nullAccept null hypothesishypothesis 11--αα (confidence(confidence levellevel(( ββ (type 2 error(type 2 error(( αα (type 1 error(type 1 error(( 11--ββ (power of the(power of the testtest(( SumSum 1.001.00 1.001.00