Research ArticleNeutrosophic DrsquoAgostino Test of Normality An Application toWater Data

Mohammed Albassam 1 Nasrullah Khan2 and Muhammad Aslam 1

1Department of Statistics Faculty of Science King Abdulaziz University Jeddah 21551 Saudi Arabia2Department of Statistics University of Veterinary and Animal Sciences Jhang Campus Lahore Pakistan

Correspondence should be addressed to Muhammad Aslam aslam_ravianhotmailcom

Received 25 January 2021 Revised 8 February 2021 Accepted 16 February 2021 Published 23 February 2021

Academic Editor Parimala Mani

Copyright copy 2021Mohammed Albassam et al(is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

(e DrsquoAgostino test has been widely applied for testing the normality of the data (e existing DrsquoAgostino test cannot be appliedwhen the data have some indeterminate observations or observations which are obtained from the complex systems In this paperwe present a DrsquoAgostino test under neutrosophic statisticsWe propose the DrsquoAgostino test to test the normality of the data havingindeterminate observations (e design of the proposed test is given and implemented with the help of real data From thecomparison it is concluded that the proposed test is effective adequate and suitable to be applied in the presenceof indeterminacy

1 Introduction

(e data obtained from various fields such as medical phys-iological education and chemical process are assumed tofollow the approximately normal distribution (erefore be-fore some estimation and forecasting the normality of the datain hand is checked first If the data follow the normal distri-bution the statistical techniques based on normal distributionare used otherwise the nonparametric methods are applied forthe analysis of the data Among many statistical tests theDrsquoAgostino test has been widely applied for testing the nor-mality of the data (is test is used to test the null hypothesisthat the data do not significantly differ from the normal dis-tribution versus the alternative hypothesis that the data sig-nificantly differ from the normality DrsquoAgostino and Stephens[1] introduced statistical tests when the data follow the normaldistribution Oztuna et al [2] studied the power of the test andtype-I error rate for various tests under normality assumptionsYap and Sim [3] discussed various statistical tests and showedthat the DrsquoAgostino test has better power Chen and Xia [4]presented tests when data are nonnormal Mishra et al [5]presented the descriptive statistic for the test More details onthe statistical test for normality can be seen in [6ndash9]

(e traditional statistical tests are applied to test the hy-pothesis that the data follow approximately normal distributionwith exact mean and variance In some situations such as themeasure of the water level a lifetime of a product and meltingof a material cannot be expressed in the exact form and haveapproximatemean and variances In this case the statistical testusing the fuzzy logic is preferable to apply for the analysis of thedata [10] Hesamian and Akbari [11] presented the tests usingfuzzy logic Chachi and Taheri [12] worked on the optimal testusing the fuzzy approach Haktanır and Kahraman [13] dis-cussed the role of tests in decision-making issues For detailsthe reader may refer to [14ndash24]

(e neutrosophic logic which is more efficient than thefuzzy logic and interval-based analysis was proposed bySmarandache [25] (is logic estimates the measures of truthfalsehood and indeterminacy while the fuzzy logic is unable toestimate the measure of indeterminacy More applications ofneutrosophic logic can be read in [26ndash36] Based on the idea ofneutrosophic logic Smarandache [37] introduced the de-scriptive neutrosophic statistics which are applied for theanalysis of the data having indeterminate observations Kan-dasamy and Smarandache [38] introduced the neutrosophicnumbers for the first time Chen et al [39] applied the

HindawiJournal of MathematicsVolume 2021 Article ID 5582102 5 pageshttpsdoiorg10115520215582102

neutrosophic numbers in rock measuring Aslam [40] intro-duced a new branch of statistical quality control under neu-trosophic statistics KolmogorovndashSmirnov tests and Bartlettand Hartley tests using neutrosophic statistics were developedby Aslam [41 42] respectively More details on the applicationof neutrosophic statistics can be seen in [43 44]

Although the DrsquoAgostino test under classical statistics isavailable in the literature the existing DrsquoAgostino testcannot be applied if observations are imprecise vague andindeterminate By exploring the literature and according tothe best of our knowledge there is work on the DrsquoAgostinotest In this paper we will propose and design the DrsquoAg-ostino test under indeterminacy (e operational process ofthe proposed test is explained (e application of the pro-posed test will be given with the help of water data Weexpect that the proposed test will be informative and

adequate than the existing DrsquoAgostino test under classicalstatistics in the indeterminate environment

2 Preliminary

Suppose that ai and biIN INε[IL IU] are determinate andindeterminate parts of neutrosophic random variablezN ai + biIN INϵ[IL IU] i 1 2 nN where nN de-notes the neutrosophic sample size (e values of zN reduceto ai when IN 0 Based on this information compute theneutrosophic average for variable zNϵ[zL zU] as follows

zN a + bIN INϵ IL IU1113858 1113859 (1)

where a (1nN) 1113936nN

i1 ai and b (1nN) 1113936nN

i1 bi(e neutrosophic sum of squares (NSS) by following

[39] is computed as follows

1113944

nN

i1zi minus ziN( 1113857

2 1113944

nN

i1

minai + biIL( 1113857 a + bIL1113872 1113873 ai + biIL( 1113857 a + bIU1113872 1113873

ai + biIU( 1113857 a + bIL1113872 1113873 ai + biIU( 1113857 a + bIU1113872 1113873

⎛⎝ ⎞⎠

maxai + biIL( 1113857 a + bIL1113872 1113873 ai + biIL( 1113857 a + bIU1113872 1113873

ai + biIU( 1113857 a + bIL1113872 1113873 ai + biIU( 1113857 a + bIU1113872 1113873

⎛⎝ ⎞⎠

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

INϵ IL IU1113858 1113859 (2)

3 Designof theProposedDrsquoAgostinoTestunderNeutrosophic Statistics

(e main objective is to design DrsquoAgostino test underneutrosophic statistics for testing the null hypothesis H0N

that the neutrosophic data follow the neutrosophic normaldistribution versus the alternative hypothesis H1N that thedata do not belong to the neutrosophic normal distribution(e acceptance of the null hypothesis means that the data arenot significantly away from the normal distribution (eoperational procedure of the proposed test is stated asfollows

Step 1 Compute the neutrosophic averages of lowervalues ai(i 1 2 nL) and upper valuesbi(i 1 2 nU) as follows a (1nN) 1113936

nN

i1 ai andb (1nN) 1113936

nN

i1 biStep 2 Find neutrosophic average as follows

zN a + bIN INϵ IL IU1113858 1113859 (3)

Step 3 (e neutrosophic sum of squares (NSS) byfollowing [39] is calculated using the followingexpression

1113944nN

i1 zi minus ziN( 11138572

1113944

nN

i1

min ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

max ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

Step 4 Compute the neutrosophic numeratorTNϵ[TL TU] of the proposed test as follows

TN 1113944 iN minusnN + 1

21113874 11138751113874 1113875XiNTNϵ TL TU1113858 1113859 (5)

where iN denotes the rank of neutrosophic observationsXiN for ai(i 1 2 nL) and bi(i 1 2 nU)Step 5 Compute the neutrosophic test statisticDNϵ[DL DU] of the proposed test as follows

DN TN

n3N 1113936

nN

i1 zi minus ziN( 11138572

1113872 1113873

1113969 TNϵ TL TU1113858 1113859 DNϵ DL DU1113858 1113859

(6)

Step 6 Decide the level of significance α and select thecritical values from the DrsquoAgostino table (e nullhypothesis will be accepted if DNϵ[DL DU] lies withinthe range of the tabulated values

2 Journal of Mathematics

4 Application for Portuguese Mineral Water

In this section we will give the application of the proposed testusing the Portuguese mineral water (PMW) data DrsquoUrso andGiordani [45] used the same data and analyzed them usingclassical statistics DrsquoUrso and Giordani [45] conidered sixmineral concentrations such as six mineral concentrations ofHCOminus

3 CIminus N+

a C2+a SiO2 and pH (e PMW data are re-

ported in Table 1 Table 1 clearly indicates that the data arereported in intervals Before any prediction or estimation isgiven for the data it is necessary to see that the data do notsignificantly differ from the normal distribution (erefore wewill apply the proposed test on these data to test whether the sixvariables are from the neutrosophic normal distribution or not

(e necessary computations for PMW data are given inthe following steps

Step 1(e neutrosophic averages of lower values ai(i

1 2 nL) and upper values bi(i 1 2 nU) ofPMW data of five different types of water are given inTable 2Step 2 (e neutrosophic averages zN INϵ[0 1]for thewater data are also shown in Table 2Step 3 (e values of NSS are given in Table 3 byfollowing [39]

1113944

nN

i1zi minus ziN( 1113857

2 1113944

nN

i1

min ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

max ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (7)

Step 4 (e values TNϵ[TL TU] and DNϵ[DL DU] arealso shown in Table 3Step 5 Let α 005 the range of the tabulated values is02513 02849 (e null hypothesis that the data followthe normal distribution is accepted if DNϵ[DL DU] iswithin the range of the tabulated values(e acceptanceor rejection of H0N is shown in Table 3 From Table 3 itis clear that the PMW data for all waters do not followthe neutrosophic normal distribution

5 Comparative Study and Discussion

(e proposed DrsquoAgostino test under neutrosophic statistics isthe extension of the DrsquoAgostino test under classical statistics

(e proposed test reduces to DrsquoAgostino test under classicalstatistics when DN DL 0 We compare the proposed testwith the existing DrsquoAgostino test using the PMW data of fivetypes of water with the same values of α (e values of statisticD for the existing test and the proposed test along with themeasure of indeterminacy are shown in Table 4 From Table 4it can be seen that the proposed test statistic DNϵ[DL DU] hasthe results in the neutrosophic form with the probability of theindeterminacy On the contrary the existing test provides onlythe determined values of statistic D For example whenα 005 and n1 the null hypothesis H0N will accepted theprobability of 095 the chance to do not acceptH0N is 005 andthe probability of indeterminacy is 00621 From the proposedtest it can be seen that 095+005+0062gt 1 which shows thecase of paraconsistent neutrosophic probability see [37]

Table 1 (e PMW data

Portuguese mineraln 1 n 2 n 3 n 4 n 5

ai bi ai bi ai bi ai bi ai bi

HCOminus3 21 41 113 119 22 42 8 116 46 5

CIminus 7 9 165 175 36 4 41 47 66 74N+

a 10 16 103 107 28 38 28 36 54 56C2+

a 3 4 15 21 001 101 19 29 072 084SiO2 23 29 137 149 101 78 58 68 167 183pH 61 65 67 71 571 581 59 6 54 58

Table 2 Neutrosophic means of five different types of water

Water aN bN zN

n 1 1168 1758 [1168 2926]n 2 292 317 [292 609]n 3 255 443 [255 698]n 4 475 593 [475 1068]n 5 557 715 [557 1272]

Journal of Mathematics 3

On the contrary the existing test provides only the determinedvalue which is not adequate when the data have intervaluncertain and indeterminate values or the data are obtainedfrom the complex system From this comparison it is con-cluded that the proposed test provides the values of statistic inthe indeterminate interval and this theory is the same as in[39] (erefore the use of the proposed test is adequate underan indeterminate environment

6 Concluding Remarks

In this paper we presented a DrsquoAgostino test under neu-trosophic statistics We proposed the DrsquoAgostino test totest the normality of the data having indeterminate ob-servations (e design of the proposed test was given andimplemented with the help of real data (e proposed testwas the extension of an existing DrsquoAgostino test underclassical statistics From the comparison it was concludedthat the proposed test is effective adequate and suitable tobe applied in the presence of indeterminacy (e devel-opment of software for the proposed test will be a fruitfularea of research(e application of the proposed test for bigdatasets such as testing the normality of ocean dataFacebook user data and rail data can be considered asfuture research

Data Availability

(e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is article was funded by the Deanship of Scientific Re-search (DSR) King Abdulaziz University Jeddah (Grant no

D-103-130ndash1441) (e authors therefore gratefully ac-knowledge DSR for the technical and financial support

References

[1] R DrsquoAgostino and M Stephens Tests for Normal Distributionin Goodness-Of-Fit Techniques Statistics Textbooks andMonographs R B DrsquoAgostino and Stephens Eds DekkerNew York NY USA 1986

[2] D Oztuna A H Elhan and E Tuccar ldquoInvestigation of fourdifferent normality tests in terms of type 1 error rate andpower under different distributionsrdquo Turkish Journal ofMedical Sciences vol 36 no 3 pp 171ndash176 2006

[3] B W Yap and C H Sim ldquoComparisons of various types ofnormality testsrdquo Journal of Statistical Computation andSimulation vol 81 no 12 pp 2141ndash2155 2011

[4] H Chen and Y Xia ldquoA nonparametric normality test forhigh-dimensional datardquo 2019 httpsarxivorgpdf190405289pdf

[5] P Mishra C M Pandey U Singh A Gupta C Sahu andA Keshri ldquoDescriptive statistics and normality tests forstatistical datardquo Annals of Cardiac Anaesthesia vol 22 no 1p 67 2019

[6] E S Pearson R B DrsquorsquorsquoAgostino and K O Bowman ldquoTestsfor departure from normality comparison of powersrdquo Bio-metrika vol 64 no 2 pp 231ndash246 1977

[7] R B Drsquoagostino A Belanger and R B DrsquoAgostino Jr ldquoAsuggestion for using powerful and informative tests of nor-malityrdquo3e American Statistician vol 44 no 4 pp 316ndash3211990

[8] L Baringhaus and N Henze ldquoA test for uniformity withunknown limits based on drsquoagostinorsquos Drdquo Statistics ampProbability Letters vol 9 no 4 pp 299ndash304 1990

[9] S Kallithraka I S Arvanitoyannis P Kefalas A El-ZajouliE Soufleros and E Psarra ldquoInstrumental and sensory analysisof Greek wines implementation of principal componentanalysis (PCA) for classification according to geographicaloriginrdquo Food Chemistry vol 73 no 4 pp 501ndash514 2001

[10] NWatanabe and T Imaizumi ldquoA fuzzy statistical test of fuzzyhypothesesrdquo Fuzzy Sets and Systems vol 53 no 2pp 167ndash178 1993

Table 4 (e comparison of two tests

Water(e proposed test (e existing test

DNϵ[DL DU] Measure of indeterminacy INϵ[IL LU] D

n 1 [01764 01142] 01764ndash01142 IN [0 00621] 01764n 2 [02104 01107] 02104ndash01107 IN [0 090] 02104n 3 [07847 06489] 07847ndash06489 IN [0 021] 07847n 4 [01533 00810] 01533ndash00810 IN [0 0892] 01533n 5 [01489 00784] 01489ndash00784 IN [0 00899] 01489

Table 3 (e values of NSS of five waters

Water NSS TNϵ[TL TU] DNϵ[DL DU] Decision

n 1 [81101 491523] [7385 11775] [01764 01142] Do not accept H0N

n 2 [785836 3317661] [2742 2965] [02104 01107] Do not accept H0N

n 3 [5455 26879] [1843 2009] [07847 06489] Do not accept H0N

n 4 [8473 52131] [2075 272] [01533 00810] Do not accept H0N

n 5 [38516 168690] [4295 4735] [01489 00784] Do not accept H0N

4 Journal of Mathematics

4 Application for Portuguese Mineral Water

In this section we will give the application of the proposed testusing the Portuguese mineral water (PMW) data DrsquoUrso andGiordani [45] used the same data and analyzed them usingclassical statistics DrsquoUrso and Giordani [45] conidered sixmineral concentrations such as six mineral concentrations ofHCOminus

3 CIminus N+

a C2+a SiO2 and pH (e PMW data are re-

ported in Table 1 Table 1 clearly indicates that the data arereported in intervals Before any prediction or estimation isgiven for the data it is necessary to see that the data do notsignificantly differ from the normal distribution (erefore wewill apply the proposed test on these data to test whether the sixvariables are from the neutrosophic normal distribution or not

(e necessary computations for PMW data are given inthe following steps

Step 1(e neutrosophic averages of lower values ai(i

1 2 nL) and upper values bi(i 1 2 nU) ofPMW data of five different types of water are given inTable 2Step 2 (e neutrosophic averages zN INϵ[0 1]for thewater data are also shown in Table 2Step 3 (e values of NSS are given in Table 3 byfollowing [39]

1113944

nN

i1zi minus ziN( 1113857

2 1113944

nN

i1

min ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

max ai minus a( 11138572 ai minus a( 1113857 ai minus a( 1113857( + 1 times bi minus b1113872 11138731113872 1113873 ai minus a( 1113857 + 1 times bi minus b1113872 1113873

21113874 11138751113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (7)

Step 4 (e values TNϵ[TL TU] and DNϵ[DL DU] arealso shown in Table 3Step 5 Let α 005 the range of the tabulated values is02513 02849 (e null hypothesis that the data followthe normal distribution is accepted if DNϵ[DL DU] iswithin the range of the tabulated values(e acceptanceor rejection of H0N is shown in Table 3 From Table 3 itis clear that the PMW data for all waters do not followthe neutrosophic normal distribution

5 Comparative Study and Discussion

(e proposed DrsquoAgostino test under neutrosophic statistics isthe extension of the DrsquoAgostino test under classical statistics

(e proposed test reduces to DrsquoAgostino test under classicalstatistics when DN DL 0 We compare the proposed testwith the existing DrsquoAgostino test using the PMW data of fivetypes of water with the same values of α (e values of statisticD for the existing test and the proposed test along with themeasure of indeterminacy are shown in Table 4 From Table 4it can be seen that the proposed test statistic DNϵ[DL DU] hasthe results in the neutrosophic form with the probability of theindeterminacy On the contrary the existing test provides onlythe determined values of statistic D For example whenα 005 and n1 the null hypothesis H0N will accepted theprobability of 095 the chance to do not acceptH0N is 005 andthe probability of indeterminacy is 00621 From the proposedtest it can be seen that 095+005+0062gt 1 which shows thecase of paraconsistent neutrosophic probability see [37]

Table 1 (e PMW data

Portuguese mineraln 1 n 2 n 3 n 4 n 5

ai bi ai bi ai bi ai bi ai bi

HCOminus3 21 41 113 119 22 42 8 116 46 5

CIminus 7 9 165 175 36 4 41 47 66 74N+

a 10 16 103 107 28 38 28 36 54 56C2+

a 3 4 15 21 001 101 19 29 072 084SiO2 23 29 137 149 101 78 58 68 167 183pH 61 65 67 71 571 581 59 6 54 58

Table 2 Neutrosophic means of five different types of water

Water aN bN zN

n 1 1168 1758 [1168 2926]n 2 292 317 [292 609]n 3 255 443 [255 698]n 4 475 593 [475 1068]n 5 557 715 [557 1272]

Journal of Mathematics 3

On the contrary the existing test provides only the determinedvalue which is not adequate when the data have intervaluncertain and indeterminate values or the data are obtainedfrom the complex system From this comparison it is con-cluded that the proposed test provides the values of statistic inthe indeterminate interval and this theory is the same as in[39] (erefore the use of the proposed test is adequate underan indeterminate environment

6 Concluding Remarks

In this paper we presented a DrsquoAgostino test under neu-trosophic statistics We proposed the DrsquoAgostino test totest the normality of the data having indeterminate ob-servations (e design of the proposed test was given andimplemented with the help of real data (e proposed testwas the extension of an existing DrsquoAgostino test underclassical statistics From the comparison it was concludedthat the proposed test is effective adequate and suitable tobe applied in the presence of indeterminacy (e devel-opment of software for the proposed test will be a fruitfularea of research(e application of the proposed test for bigdatasets such as testing the normality of ocean dataFacebook user data and rail data can be considered asfuture research

Data Availability

(e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is article was funded by the Deanship of Scientific Re-search (DSR) King Abdulaziz University Jeddah (Grant no

D-103-130ndash1441) (e authors therefore gratefully ac-knowledge DSR for the technical and financial support

References

[1] R DrsquoAgostino and M Stephens Tests for Normal Distributionin Goodness-Of-Fit Techniques Statistics Textbooks andMonographs R B DrsquoAgostino and Stephens Eds DekkerNew York NY USA 1986

[2] D Oztuna A H Elhan and E Tuccar ldquoInvestigation of fourdifferent normality tests in terms of type 1 error rate andpower under different distributionsrdquo Turkish Journal ofMedical Sciences vol 36 no 3 pp 171ndash176 2006

[3] B W Yap and C H Sim ldquoComparisons of various types ofnormality testsrdquo Journal of Statistical Computation andSimulation vol 81 no 12 pp 2141ndash2155 2011

[4] H Chen and Y Xia ldquoA nonparametric normality test forhigh-dimensional datardquo 2019 httpsarxivorgpdf190405289pdf

[5] P Mishra C M Pandey U Singh A Gupta C Sahu andA Keshri ldquoDescriptive statistics and normality tests forstatistical datardquo Annals of Cardiac Anaesthesia vol 22 no 1p 67 2019

[6] E S Pearson R B DrsquorsquorsquoAgostino and K O Bowman ldquoTestsfor departure from normality comparison of powersrdquo Bio-metrika vol 64 no 2 pp 231ndash246 1977

[7] R B Drsquoagostino A Belanger and R B DrsquoAgostino Jr ldquoAsuggestion for using powerful and informative tests of nor-malityrdquo3e American Statistician vol 44 no 4 pp 316ndash3211990

[8] L Baringhaus and N Henze ldquoA test for uniformity withunknown limits based on drsquoagostinorsquos Drdquo Statistics ampProbability Letters vol 9 no 4 pp 299ndash304 1990

[9] S Kallithraka I S Arvanitoyannis P Kefalas A El-ZajouliE Soufleros and E Psarra ldquoInstrumental and sensory analysisof Greek wines implementation of principal componentanalysis (PCA) for classification according to geographicaloriginrdquo Food Chemistry vol 73 no 4 pp 501ndash514 2001

[10] NWatanabe and T Imaizumi ldquoA fuzzy statistical test of fuzzyhypothesesrdquo Fuzzy Sets and Systems vol 53 no 2pp 167ndash178 1993

Table 4 (e comparison of two tests

Water(e proposed test (e existing test

DNϵ[DL DU] Measure of indeterminacy INϵ[IL LU] D

n 1 [01764 01142] 01764ndash01142 IN [0 00621] 01764n 2 [02104 01107] 02104ndash01107 IN [0 090] 02104n 3 [07847 06489] 07847ndash06489 IN [0 021] 07847n 4 [01533 00810] 01533ndash00810 IN [0 0892] 01533n 5 [01489 00784] 01489ndash00784 IN [0 00899] 01489

Table 3 (e values of NSS of five waters

Water NSS TNϵ[TL TU] DNϵ[DL DU] Decision

n 1 [81101 491523] [7385 11775] [01764 01142] Do not accept H0N

n 2 [785836 3317661] [2742 2965] [02104 01107] Do not accept H0N

n 3 [5455 26879] [1843 2009] [07847 06489] Do not accept H0N

n 4 [8473 52131] [2075 272] [01533 00810] Do not accept H0N

n 5 [38516 168690] [4295 4735] [01489 00784] Do not accept H0N

4 Journal of Mathematics

On the contrary the existing test provides only the determinedvalue which is not adequate when the data have intervaluncertain and indeterminate values or the data are obtainedfrom the complex system From this comparison it is con-cluded that the proposed test provides the values of statistic inthe indeterminate interval and this theory is the same as in[39] (erefore the use of the proposed test is adequate underan indeterminate environment

6 Concluding Remarks

In this paper we presented a DrsquoAgostino test under neu-trosophic statistics We proposed the DrsquoAgostino test totest the normality of the data having indeterminate ob-servations (e design of the proposed test was given andimplemented with the help of real data (e proposed testwas the extension of an existing DrsquoAgostino test underclassical statistics From the comparison it was concludedthat the proposed test is effective adequate and suitable tobe applied in the presence of indeterminacy (e devel-opment of software for the proposed test will be a fruitfularea of research(e application of the proposed test for bigdatasets such as testing the normality of ocean dataFacebook user data and rail data can be considered asfuture research

Data Availability

(e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is article was funded by the Deanship of Scientific Re-search (DSR) King Abdulaziz University Jeddah (Grant no

D-103-130ndash1441) (e authors therefore gratefully ac-knowledge DSR for the technical and financial support

References

[1] R DrsquoAgostino and M Stephens Tests for Normal Distributionin Goodness-Of-Fit Techniques Statistics Textbooks andMonographs R B DrsquoAgostino and Stephens Eds DekkerNew York NY USA 1986

[2] D Oztuna A H Elhan and E Tuccar ldquoInvestigation of fourdifferent normality tests in terms of type 1 error rate andpower under different distributionsrdquo Turkish Journal ofMedical Sciences vol 36 no 3 pp 171ndash176 2006

[3] B W Yap and C H Sim ldquoComparisons of various types ofnormality testsrdquo Journal of Statistical Computation andSimulation vol 81 no 12 pp 2141ndash2155 2011

[4] H Chen and Y Xia ldquoA nonparametric normality test forhigh-dimensional datardquo 2019 httpsarxivorgpdf190405289pdf

[5] P Mishra C M Pandey U Singh A Gupta C Sahu andA Keshri ldquoDescriptive statistics and normality tests forstatistical datardquo Annals of Cardiac Anaesthesia vol 22 no 1p 67 2019

[6] E S Pearson R B DrsquorsquorsquoAgostino and K O Bowman ldquoTestsfor departure from normality comparison of powersrdquo Bio-metrika vol 64 no 2 pp 231ndash246 1977

[7] R B Drsquoagostino A Belanger and R B DrsquoAgostino Jr ldquoAsuggestion for using powerful and informative tests of nor-malityrdquo3e American Statistician vol 44 no 4 pp 316ndash3211990

[8] L Baringhaus and N Henze ldquoA test for uniformity withunknown limits based on drsquoagostinorsquos Drdquo Statistics ampProbability Letters vol 9 no 4 pp 299ndash304 1990

[9] S Kallithraka I S Arvanitoyannis P Kefalas A El-ZajouliE Soufleros and E Psarra ldquoInstrumental and sensory analysisof Greek wines implementation of principal componentanalysis (PCA) for classification according to geographicaloriginrdquo Food Chemistry vol 73 no 4 pp 501ndash514 2001

[10] NWatanabe and T Imaizumi ldquoA fuzzy statistical test of fuzzyhypothesesrdquo Fuzzy Sets and Systems vol 53 no 2pp 167ndash178 1993

Table 4 (e comparison of two tests

Water(e proposed test (e existing test

DNϵ[DL DU] Measure of indeterminacy INϵ[IL LU] D

n 1 [01764 01142] 01764ndash01142 IN [0 00621] 01764n 2 [02104 01107] 02104ndash01107 IN [0 090] 02104n 3 [07847 06489] 07847ndash06489 IN [0 021] 07847n 4 [01533 00810] 01533ndash00810 IN [0 0892] 01533n 5 [01489 00784] 01489ndash00784 IN [0 00899] 01489

Table 3 (e values of NSS of five waters

Water NSS TNϵ[TL TU] DNϵ[DL DU] Decision

n 1 [81101 491523] [7385 11775] [01764 01142] Do not accept H0N

n 2 [785836 3317661] [2742 2965] [02104 01107] Do not accept H0N

n 3 [5455 26879] [1843 2009] [07847 06489] Do not accept H0N

n 4 [8473 52131] [2075 272] [01533 00810] Do not accept H0N

n 5 [38516 168690] [4295 4735] [01489 00784] Do not accept H0N

4 Journal of Mathematics

[11] G Hesamian and M G Akbari ldquoStatistical test based onintuitionistic fuzzy hypothesesrdquo Communications in Statis-tics-3eory and Methods vol 46 no 18 pp 9324ndash9334 2017

[12] J Chachi and S M Taheri ldquoOptimal statistical tests based onfuzzy random variablesrdquo Iranian Journal of Fuzzy Systemsvol 15 no 5 pp 27ndash45 2018

[13] E Haktanır and C Kahraman ldquoFuzzy hypothesis testing instatistical decision makingrdquo Journal of Intelligent amp FuzzySystems (Preprint) vol 37 no 5 pp 6545ndash6555 2019

[14] B Van Cutsem and I Gath ldquoDetection of outliers and robustestimation using fuzzy clusteringrdquo Computational Statistics ampData Analysis vol 15 no 1 pp 47ndash61 1993

[15] M Montenegro M A R Casals M A A Lubiano andM A A Gil ldquoTwo-sample hypothesis tests of means of a fuzzyrandom variablerdquo Information Sciences vol 133 no 1-2pp 89ndash100 2001

[16] V Mohanty and P AnnanNaidu ldquoFraud detection usingoutlier analysis a surveyrdquo International Journal of Engi-neering Sciences and Research Technology vol 2 no 6 2013

[17] Y M Moradnezhadi ldquoDetermination of a some simplemethods for outlier detection in maximum daily rainfall (casestudy baliglichay Watershed BasinndashArdebil ProvincendashIran)rdquoBulletin of Environment Pharmacology and Life Sciencesvol 3 no 3 pp 110ndash117 2014

[18] T-Y Ling C-L Soo J-J Liew L Nyanti S-F Sim andJ Grinang ldquoApplication of multivariate statistical analysis inevaluation of surface river water quality of a tropical riverrdquoJournal of Chemistry vol 2017 2017

[19] G Shahzadi M Akram and A B Saeid ldquoAn application ofsingle-valued neutrosophic sets in medical diagnosisrdquo Neu-trosophic Sets and Systems vol 18 pp 80ndash88 2017

[20] A K Shrestha and N Basnet ldquo(e Correlation and regressionanalysis of physicochemical parameters of River water for theevaluation of percentage contribution to electrical conduc-tivityrdquo Journal of Chemistry vol 2018 2018

[21] Y Choi H Lee and Z Irani ldquoBig data-driven fuzzy cognitivemap for prioritising IT service procurement in the publicsectorrdquo Annals of Operations Research vol 270 no 1-2pp 75ndash104 2018

[22] S Habib and M Akram ldquoDiagnostic methods and riskanalysis based on fuzzy soft informationrdquo InternationalJournal of Biomathematics vol 11 no 08 Article ID 18500962018

[23] S Habib and M Akram ldquoMedical decision support systemsbased on Fuzzy Cognitive Mapsrdquo International Journal ofBiomathematics vol 12 no 06 Article ID 1950069 2019

[24] S Habib M A Butt M Akram and F Smarandache ldquoAneutrosophic clinical decision-making system for cardiovas-cular diseases risk analysisrdquo Journal of Intelligent amp FuzzySystems (Preprint) vol 39 no 5 pp 7807ndash7829 2020

[25] F Smarandache Neutrosophy Neutrosophic Probability Setand Logic vol 105 pp 118ndash123 ProQuest Information ampLearning Ann Arbor MI USA 1998

[26] HWang F Smarandache R Sunderraman and Y-Q ZhangldquoInterval neutrosophic sets and logic theory and applicationsin computing theory and applications in computingrdquo InfiniteStudy vol 5 2005

[27] I Hanafy A Salama and KMahfouz ldquoNeutrosophic classicalevents and its probabilityrdquo International Journal of Mathe-matics and Computer Applications Research (IJMCAR) vol 3no 1 pp 171ndash178 2013

[28] Y Guo and A Sengur ldquoNECM neutrosophic evidentialc-means clustering algorithmrdquo Neural Computing and Ap-plications vol 26 no 3 pp 561ndash571 2015

[29] M Abdel-Basset G Manogaran A Gamal andF Smarandache ldquoA hybrid approach of neutrosophic sets andDEMATEL method for developing supplier selection crite-riardquo Design Automation for Embedded Systems vol 22 no 3pp 257ndash278 2018

[30] R Alhabib M M Ranna H Farah and A Salama ldquoSomeneutrosophic probability distributionsrdquo Neutrosophic Setsand Systems vol 30 2018

[31] S Broumi A Bakali M Talea and F Smarandache ldquoBipolarneutrosophic minimum spanning treerdquo Infinite Studyvol 127 2018

[32] X Peng and J Dai ldquoApproaches to single-valued neu-trosophic MADM based on MABAC TOPSIS and newsimilarity measure with score functionrdquo Neural Computingand Applications vol 29 no 10 pp 939ndash954 2018

[33] A I Shahin K M Amin A A Sharawi and Y Guo ldquoA novelenhancement technique for pathological microscopic imageusing neutrosophic similarity score scalingrdquo Optik vol 161pp 84ndash97 2018

[34] M Abdel-Basset M Mohamed M Elhoseny L H SonF Chiclana and A E-N H Zaied ldquoCosine similarity mea-sures of bipolar neutrosophic set for diagnosis of bipolardisorder diseasesrdquo Artificial Intelligence in Medicine vol 101Article ID 101735 2019

[35] C Jana and M Pal ldquoA robust single-valued neutrosophic softaggregation operators in multi-criteria decision makingrdquoSymmetry vol 11 no 1 Article ID 110 2019

[36] N A Nabeeh M Abdel-Basset H A El-Ghareeb andA Aboelfetouh ldquoNeutrosophic multi-criteria decision mak-ing approach for iot-based enterprisesrdquo Institute of Electricaland Electronics Engineers Access vol 7 pp 59559ndash595742019

[37] F Smarandache ldquoIntroduction to neutrosophic statisticsrdquoInfinite Study httpsarxivorgabs14062000 2014

[38] W V Kandasamy and F Smarandache ldquoFuzzy cognitivemaps and neutrosophic cognitive mapsrdquo Infinite Studyhttpsarxivorgabsmath0311063 2003

[39] J Chen J Ye and S Du ldquoScale effect and anisotropy analyzedfor neutrosophic numbers of rock joint roughness coefficientbased on neutrosophic statisticsrdquo Symmetry vol 9 no 10p 208 2017

[40] M Aslam ldquoA new sampling plan using neutrosophic processloss considerationrdquo Symmetry vol 10 no 5 p 132 2018

[41] M Aslam ldquoIntroducing Kolmogorovndashsmirnov tests underuncertainty an application to radioactive datardquo ACS Omegavol 5 no 1 pp 914ndash917 2019

[42] M Aslam ldquoDesign of the Bartlett and Hartley tests for ho-mogeneity of variances under indeterminacy environmentrdquoJournal of Taibah University for Science vol 14 no 1pp 6ndash10 2020

[43] M Aslam ldquoNeutrosophic analysis of variance application touniversity studentsrdquo Complex amp Intelligent Systems vol 5no 4 pp 403ndash407 2019

[44] M Aslam and M Albassam ldquoApplication of neutrosophiclogic to evaluate correlation between prostate cancer mortalityand dietary fat assumptionrdquo Symmetry vol 11 no 3 p 3302019

[45] P DrsquoUrso and P Giordani ldquoA least squares approach toprincipal component analysis for interval valued datardquoChemometrics and Intelligent Laboratory Systems vol 70no 2 pp 179ndash192 2004

[46] J Chen J Ye S Du and R Yong ldquoExpressions of rock jointroughness coefficient using neutrosophic interval statisticalnumbersrdquo Symmetry vol 9 no 7 p 123 2017

Journal of Mathematics 5