# Neutrosophic D’Agostino Test of Normality: An Application to Water Data

Authors: Mohammed Albassam; Nasrullah Khan; Muhammad Aslam
Journal: Journal of Mathematics
 (2021)

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## Abstract

The D’Agostino test has been widely applied for testing the normality of the data. The existing D’Agostino test cannot be applied when the data have some indeterminate observations or observations which are obtained from the complex systems. In this paper, we present a D’Agostino test under neutrosophic statistics. We propose the D’Agostino test to test the normality of the data having indeterminate observations. The design of the proposed test is given and implemented with the help of real data. From the comparison, it is concluded that the proposed test is effective, adequate, and suitable to be applied in the presence of indeterminacy.


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## Body

## 1. Introduction

The data obtained from various fields such as medical, physiological, education, and chemical process are assumed to follow the approximately normal distribution. Therefore, before some estimation and forecasting, the normality of the data in hand is checked first. If the data follow the normal distribution, the statistical techniques based on normal distribution are used; otherwise, the nonparametric methods are applied for the analysis of the data. Among many statistical tests, the D’Agostino test has been widely applied for testing the normality of the data. This test is used to test the null hypothesis that the data do not significantly differ from the normal distribution versus the alternative hypothesis that the data significantly differ from the normality. D’Agostino and Stephens [1] introduced statistical tests when the data follow the normal distribution. Öztuna et al. [2] studied the power of the test and type-I error rate for various tests under normality assumptions. Yap and Sim [3] discussed various statistical tests and showed that the D’Agostino 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 on the statistical test for normality can be seen in [6–9].The traditional statistical tests are applied to test the hypothesis that the data follow approximately normal distribution with exact mean and variance. In some situations, such as the measure of the water level, a lifetime of a product and melting of a material cannot be expressed in the exact form and have approximate mean and variances. In this case, the statistical test using the fuzzy logic is preferable to apply for the analysis of the data [10]. Hesamian and Akbari [11] presented the tests using fuzzy logic. Chachi and Taheri [12] worked on the optimal test using the fuzzy approach. Haktanır and Kahraman [13] discussed the role of tests in decision-making issues. For details, the reader may refer to [14–24].The neutrosophic logic which is more efficient than the fuzzy logic and interval-based analysis was proposed by Smarandache [25]. This logic estimates the measures of truth, falsehood, and indeterminacy, while the fuzzy logic is unable to estimate the measure of indeterminacy. More applications of neutrosophic logic can be read in [26–36]. Based on the idea of neutrosophic logic, Smarandache [37] introduced the descriptive neutrosophic statistics which are applied for the analysis of the data having indeterminate observations. Kandasamy and Smarandache [38] introduced the neutrosophic numbers for the first time. Chen et al. [39] applied the neutrosophic numbers in rock measuring. Aslam [40] introduced a new branch of statistical quality control under neutrosophic statistics. Kolmogorov–Smirnov tests and Bartlett and Hartley tests using neutrosophic statistics were developed by Aslam [41, 42], respectively. More details on the application of neutrosophic statistics can be seen in [43, 44].Although the D’Agostino test under classical statistics is available in the literature, the existing D’Agostino test cannot be applied if observations are imprecise, vague, and indeterminate. By exploring the literature and according to the best of our knowledge, there is work on the D’Agostino test. In this paper, we will propose and design the D’Agostino test under indeterminacy. The operational process of the proposed test is explained. The application of the proposed test will be given with the help of water data. We expect that the proposed test will be informative and adequate than the existing D’Agostino test under classical statistics in the indeterminate environment.


## 2. Preliminary

Suppose thatai and biIN;INεIL,IU are determinate and indeterminate parts of neutrosophic random variable zN=ai+biIN;INϵIL,IU, i=1,2,…,nN, where nN denotes the neutrosophic sample size. The values of zN reduce to ai when IN=0. Based on this information, compute the neutrosophic average for variable zNϵzL,zU as follows:(1)z¯N=a¯+b¯IN,INϵIL,IU,where a¯=1/nN∑i=1nNai and b¯=1/nN∑i=1nNbi.The neutrosophic sum of squares (NSS) by following [39] is computed as follows:(2)∑i=1nNzi−z¯iN2=∑i=1nNminai+biILa¯+b¯IL,ai+biILa¯+b¯IUai+biIUa¯+b¯IL,ai+biIUa¯+b¯IUmaxai+biILa¯+b¯IL,ai+biILa¯+b¯IUai+biIUa¯+b¯IL,ai+biIUa¯+b¯IU,INϵIL,IU.


## 3. Design of the Proposed D’Agostino Test under Neutrosophic Statistics

The main objective is to design D’Agostino test under neutrosophic statistics for testing the null hypothesisH0N that the neutrosophic data follow the neutrosophic normal distribution versus the alternative hypothesis H1N that the data do not belong to the neutrosophic normal distribution. The acceptance of the null hypothesis means that the data are not significantly away from the normal distribution. The operational procedure of the proposed test is stated as follows.Step 1: Compute the neutrosophic averages of lower valuesaii=1,2,…,nL and upper values bii=1,2,…,nU as follows: a¯=1/nN∑i=1nNai and. b¯=1/nN∑i=1nNbi.Step 2: Find neutrosophic average as follows:(3)z¯N=a¯+b¯IN,INϵIL,IU.Step 3: The neutrosophic sum of squares (NSS) by following [39] is calculated using the following expression:(4)∑i=1nNzi−z¯iN2=∑i=1nNminai−a¯2,ai−a¯ai−a¯+1×bi−b¯,ai−a¯+1×bi−b¯2maxai−a¯2,ai−a¯ai−a¯+1×bi−b¯,ai−a¯+1×bi−b¯2.Step 4: Compute the neutrosophic numeratorTNϵTL,TU of the proposed test as follows:(5)TN=∑iN−nN+12XiNTNϵTL,TU,where iN denotes the rank of neutrosophic observations XiN for aii=1,2,…,nL and bii=1,2,…,nU.Step 5: Compute the neutrosophic test statisticDNϵDL,DU of the proposed test as follows:(6)DN=TNnN3∑i=1nNzi−z¯iN2,TNϵTL,TU,DNϵDL,DU.Step 6: Decide the level of significanceα and select the critical values from the D’Agostino table. The null hypothesis will be accepted if DNϵDL,DU lies within the range of the tabulated values.


## 4. Application for Portuguese Mineral Water

In this section, we will give the application of the proposed test using the Portuguese mineral water (PMW) data. D’Urso and Giordani [45] used the same data and analyzed them using classical statistics. D’Urso and Giordani [45] conidered six mineral concentrations such as six mineral concentrations of HCO3−, CI−, Na+, Ca2+, SiO2, and pH. The PMW data are reported in Table 1. Table 1 clearly indicates that the data are reported in intervals. Before any prediction or estimation is given for the data, it is necessary to see that the data do not significantly differ from the normal distribution. Therefore, we will apply the proposed test on these data to test whether the six variables are from the neutrosophic normal distribution or not.Table 1
The PMW data.
Portuguese mineraln. 1n. 2n. 3n. 4n. 5aibiaibiaibiaibiaibiHCO3−21411131192.24.2811.64.65CI−7916.517.53.644.14.76.67.4Na+101610.310.72.83.82.83.65.45.6Ca2+3415210.011.011.92.90.720.84SiO2232913.714.91.017.85.86.816.718.3pH6.16.56.77.15.715.815.965.45.8The necessary computations for PMW data are given in the following steps.Step 1: The neutrosophic averages of lower valuesaii=1,2,…,nL and upper values bii=1,2,…,nU of PMW data of five different types of water are given in Table 2.Step 2: The neutrosophic averagesz¯N;INϵ0,1for the water data are also shown in Table 2.Step 3: The values of NSS are given in Table3 by following [39]:(7)∑i=1nNzi−z¯iN2=∑i=1nNminai−a¯2,ai−a¯ai−a¯+1×bi−b¯,ai−a¯+1×bi−b¯2maxai−a¯2,ai−a¯ai−a¯+1×bi−b¯,ai−a¯+1×bi−b¯2.Step 4: The valuesTNϵTL,TU and DNϵDL,DU are also shown in Table 3.Step 5: Letα=0.05; the range of the tabulated values is 0.2513, 0.2849. The null hypothesis that the data follow the normal distribution is accepted if DNϵDL,DU is within the range of the tabulated values. The acceptance or rejection of H0N is shown in Table 3. From Table 3, it is clear that the PMW data for all waters do not follow the neutrosophic normal distribution.Table 2
Neutrosophic means of five different types of water.
Watera¯Nb¯Nz¯Nn. 111.6817.58[11.68, 29.26]n. 229.231.7[29.2, 60.9]n. 32.554.43[2.55, 6.98]n. 44.755.93[4.75, 10.68]n. 55.577.15[5.57, 12.72]Table 3
The values of NSS of five waters.
WaterNSSTNϵTL,TUDNϵDL,DUDecisionn. 1[811.01, 4915.23][73.85, 117.75][0.1764, 0.1142]Do not acceptH0Nn. 2[7858.36, 33176.61][274.2, 296.5][0.2104, 0.1107]Do not acceptH0Nn. 3[54.55, 268.79][18.43, 20.09][0.7847, 0.6489]Do not acceptH0Nn. 4[84.73, 521.31][20.75, 27.2][0.1533, 0.0810]Do not acceptH0Nn. 5[385.16, 1686.90][42.95, 47.35][0.1489, 0.0784]Do not acceptH0N


## 5. Comparative Study and Discussion

The proposed D’Agostino test under neutrosophic statistics is the extension of the D’Agostino test under classical statistics. The proposed test reduces to D’Agostino test under classical statistics whenDN=DL=0. We compare the proposed test with the existing D’Agostino test using the PMW data of five types of water with the same values of α. The values of statistic D for the existing test and the proposed test along with the measure of indeterminacy are shown in Table 4. From Table 4, it can be seen that the proposed test statistic DNϵDL,DU has the results in the neutrosophic form with the probability of the indeterminacy. On the contrary, the existing test provides only the determined values of statistic D. For example, when α = 0.05 and n.1, the null hypothesis H0N will accepted the probability of 0.95, the chance to do not accept H0N is 0.05, and the probability of indeterminacy is 0.0621. From the proposed test, it can be seen that 0.95 + 0.05 + 0.062 > 1 which shows the case of paraconsistent neutrosophic probability, see [37]. On the contrary, the existing test provides only the determined value which is not adequate when the data have interval, uncertain, and indeterminate values or the data are obtained from the complex system. From this comparison, it is concluded that the proposed test provides the values of statistic in the indeterminate interval, and this theory is the same as in [39]. Therefore, the use of the proposed test is adequate under an indeterminate environment.Table 4
The comparison of two tests.
WaterThe proposed testThe existing testDNϵDL,DUMeasure of indeterminacyINϵIL,LUDn. 1[0.1764, 0.1142]0.1764–0.1142IN; [0, 0.0621]0.1764n. 2[0.2104, 0.1107]0.2104–0.1107IN; [0, 0.90]0.2104n. 3[0.7847, 0.6489]0.7847–0.6489IN; [0, 0.21]0.7847n. 4[0.1533, 0.0810]0.1533–0.0810IN; [0, 0.892]0.1533n. 5[0.1489, 0.0784]0.1489–0.0784IN; [0, 0.0.899]0.1489


## 6. Concluding Remarks

In this paper, we presented a D’Agostino test under neutrosophic statistics. We proposed the D’Agostino test to test the normality of the data having indeterminate observations. The design of the proposed test was given and implemented with the help of real data. The proposed test was the extension of an existing D’Agostino test under classical statistics. From the comparison, it was concluded that the proposed test is effective, adequate, and suitable to be applied in the presence of indeterminacy. The development of software for the proposed test will be a fruitful area of research. The application of the proposed test for big datasets such as testing the normality of ocean data, Facebook user data, and rail data can be considered as future research.



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*Source: 5582102-2021-02-23.xml*