Examination of Proper Randomness of the Numbers Generated by Rand Corporation (1955) Random Number Table: t-test

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1 ISSN(Olie) :- ISSN (Prit) : - (A ISO : Certified Orgaizatio) Vol., Issue, October Examiatio of Proper Radomess of the Numbers Geerated by Rad Corporatio () Radom Number Table: t-test Brajedra Kata Sarmah, Dhritikesh Chakrabarty, Associate Professor, Departmet of Statistics, Bholaath College, Dhubri, Assam, Idia Associate Professor, Departmet of Statistics, Hadique Girls College, Assam, Idia ABSTRACT: Proper radomess of the umbers geerated by Rad Corporatio has bee tested by B.K. Sarmah, D. Chakrabarty ad N. Barma, IJESM () : Jauary March,, - by applyig the Chi-square test for testig the sigificace of differece betwee observed frequecy of each of the digit i the table ad the correspodig theoretical (expected) frequecy. I this paper, the radomess of the digits have bee tested by applyig t-test for amout of deviatio of the observed umber of occurreces ad the theoretical (expected) umber of occurreces of the respective digits ad hece the umbers. The test shows that the umbers geerated by Rad Corporatio deviated sigificatly i most the observatios from proper radomess. KEYWORDS: Radom umber geerated by Rad Corporatio, studet t-test, testig of radomess. I. INTRODUCTION Drawig of radom sample has bee foud to be vital or basic ecessity i most of the researches ad ivestigatios especially of applied scieces. The coveiet practical method of selectig a radom sample cosists of the use of Table of Radom Numbers. Existig tables of radom umbers, used commoly, are the oes due to Fisher ad Yates (Costructed i ), L.H.C. Tippett (Costructed i ), Kedall ad Babigto Smith (Costructed i ) ad Rad Corporatio (costructed i ) The radom umber tables have bee subjected to various statistical tests of radomess. These tests have limitatios to decide o proper radomess of the umbers occurrig i the correspodig tables. As a Cosequece it is ot guarateed that the umbers i each of these tables are properly radom. This leads to thik of testig the proper radomess of the umbers i the tables. I the preset study, a attempt has bee made to test this. The study, here, has bee made o the testig of radomess of the table of umbers costructed by Rad Corporatio oly. By the existig statistical methods, it is oly possible to kow whether the radomess of the umbers of a table is proper. It is oly possible to kow whether the deviatio of the degree of its radomess is sigificat. I order to test the proper radomess of the radom umbers table costructed by Rad Corporatio, t-test has bee applied. The test shows that Rad Corporatio s Radom umbers table deviate sigificatly from proper radomess. II. MATERIALS AND METHODS Rad Corporatio radom umber table cosists of a total of oe millio radom digits grouped ito,, sets of digited radom umbers. Here a sample of digites are cosidered out of oe millio digits for testig Radomess. To kow whether the umber i radom umbers table of Rad Corporatio are proper or ot studet s t-test for amout of deviatio is applied. Let d be the variable deotig the measure of the deviatio (amout of deviatio) of the observed umber of occurreces of the respective digit. Suppose, di(i=,.) are idepedet observed values of the deviatio variables. Copyright to IJIRSET DOI:./IJIRSET..

2 ISSN(Olie) :- ISSN (Prit) : - (A ISO : Certified Orgaizatio) Vol., Issue, October If the table of umber is radom the di=, for all i, i the ideal situatio. However, due to chace error, di may assume o zero value. Thus the values of di s are due to chace error but ot due to ay assigable error if the table is radom. The chace variables are i.i.d. N.(o,σ) variables. Thus testig of radomess is equivalet to testig of the hypothesis H o That E(d i ) =, for all i, Let us cosider the statistic t for testig H o d E (d) i.e. t = t S.E.(d) where, d= We have, E (d)= Also, i= d i E d i = Whe H o is true var (d) = σ, σ is ukow However ubiased estimate of σ is s = (d i d) = [Σ d i (Σd i) ] Which implies ubiased estimate of var (d) = s ad S. D. (d) = s Therefore statistic t for testig H o becomes d t = whe H o is true ad this t follows studet s t distributio with -) d.f. s/ III. STEPS IN THE METHOD I order to test the proper radomess of the umbers of Rad Corporatio table oe is required to proceed with the followig steps: Step: I the first step, observe the occurreces of the digits to for first trails, secod trials.. up to th trails as show i the table. Step: I the secod step, compute the theoretical expected frequecies. This is doe by dividig trails i.e. st ad th trails by assumig that the digit to occurs equal umber of times. Step : I the third step, compute the amout of deviatio of observed occurreces of digits ad expected occurreces of digits. Step : I the fourth step, compute the value of studet s t for each of the trails. Step : Compare the value of t-statistic with correspodig theoretical (expected) values. Step : Draw coclusio as per the result obtaied i step. IV. RESULTS AND DISCUSSION The results obtaied o operatig the steps (Nos. to ) o the radom umbers table costructed by Rad Corporatio have bee observed. It is observed from the table that occurreces of digit to are ot equal. V. CONCLUSION From the table prepared for observed frequecy of occurrece of digits alog with respective expected frequecy (show i bracket) it is observed that the calculated value of t is sigificat i all most of the cases except Copyright to IJIRSET DOI:./IJIRSET..

3 ISSN(Olie) :- ISSN (Prit) : - (A ISO : Certified Orgaizatio) Vol., Issue, October rd, th, th, rd, th trails. That is calculated value of t have bee foud to be sigificat o comparig them with the correspodig theoretical values for most of the cases (trails). Hece it may be cocluded that the table of umbers costructed by Rad Corporatio deviates sigificatly from proper radomess. Therefore Rad Corporatio Radom Numbers Table Caot be treated as properly radom, as per results obtaied by applyig t-test. TABLE Observed frequecy of occurrece of digits alog with the respective expected frequecy (Show i bracket), amout of deviatio (di) ad the values of studets t statistic from Rad Corporatio. Digits Value of t st ( ( ( ( ( ( ( ) ) ) ) ) ) ). d rd th th th th ` th th th th th th..*....* Copyright to IJIRSET DOI:./IJIRSET..

4 ISSN(Olie) :- ISSN (Prit) : - (A ISO : Certified Orgaizatio) Vol., Issue, October th th th th th th th st d rd th th *...*..* *Idicates the values which are less tha theoretical values of t., =. REFERENCES. Cochra, W.G. (): Sample survey, Cambridge Uiversity press.. Chakrabarty, D. (): A theoretical defiitio of probability based o commo sese, Bulleti of pure ad applied scieces, E-, -.. Chakrabarty, D. (): Probability;Lik betwee classical defiitio ad Empirical defiitio, J. Ass. Sc. Soc.,,-.. Chakrabarty, D. (): No equally likely outcomes: The classical defiitio of probability, Bulleti of pure ad applied scieces,, E-, -.. Chakrabarty, D. (): Empirical defiitio of probability : Special case of its Theoretical defiatios. It. J. Agricult. Stats. Sci., (), -.. Chakrabarty, D. (): Berolli s defiitio of probability: Special case of its Chakrabarty defiitio, It. J.. Agricult. Stats. Sci., (), -.. Cochra (): A survey of Experimetal desig, Mimeo, U.S.D.A.. Fisher, R. A. (): Statistical table for biological, agricultural ad medical research, th editio (), Logma group limilted, Eglad, - & -.. Kedall, M.G. ad B. B. Simith (): Radomess ad Radom samplig umbers, Jour. Roy Sat. Soc.,, -. Copyright to IJIRSET DOI:./IJIRSET..

5 ISSN(Olie): - ISSN (Prit): - (A ISO : Certified Orgaizatio) Vol., Issue, October. Rad Corporatio (): A millio radom digits free press, Gleoe, III.. Sedecor, G.W. ad W. G. Cochra (): Statistical methods, Lowa state uiversity press, Ames, Lowa, th editio.. Tippet (). Radom umber tables tracts of computer, No.-, Cambridge Uiversity press.. Vo Mises R. (): Probability, Statistics ad Truth Macmillia.. Sarmah, B.K ad Chakrabarty, D (): Testig of Radomess of Number geerated by Fisher ad Yates (Chi-Square test), IJESRT, (), -.. Sarmah, B.K ad Chakrabarty, D (): Examiatio of Proper Radomess of the Number Geerated by L.H.C Tippett, (Chi-Square test)jesrt, (), -.. Sarmah, B.K ad Chakrabarty, D (): Examiatio of Proper Radomess of the Number Geerated by Kedall ad Babigto Smith (Chi-square test). IJESRT () Febuary/, -.. B.K. Sarmah*, D. Chakrabarty, N. Barma (): Examiatio of Proper Radomess of the Number Geerated by Rad Corporatio () (Chi-square test), IJESM (): Jauary- March,, -.. Sarmah, B.K ad Chakrabarty, D (): Testig of proper Radomess of the umbers geerated by Fisher ad Yates (Applyig t-test) ABJMI Vol- issue I, Ja- Jue, P -.. Sarmah, B.K. ad Chakrabarty D. (): Examiatio of proper Radomess of the umbers geerated by L.H.C. Tippett ()(t-test) IOSR Joural of Mathematics, Vol-, Issue- May-Jue, PP.- Copyright to IJIRSET DOI:./IJIRSET..

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of