Testing the normality of heart associated variables having neutrosophic numbers

Abstract

In this paper, tests of skewness and kurtosis are introduced under neutrosophic statistics. The necessary measures and neutrosophic forms of these estimators are introduced. The application of the proposed tests is given using the data associated with heart diseases. From the real example analysis, the proposed tests are quite flexible and informative than the existing tests under classical statistics. In addition, it is concluded from the analysis that the proposed tests give information about the measure of indeterminacy in the presence of uncertainty.

Keywords

Skewness kurtosis normality Neutrosophy heart disease

1 Introduction

The tests of kurtosis and skewness have been widely applied in social sciences and psychology where the data is usually deviated from the normality, see [1] and [2]. To see the height of the distribution from the central value, the test of kurtosis is applied. This test determines either the distribution is leptokurtic distribution, mesokurtic (normal), and platykurtic. On the other hand, the test of skewness is applied to see either the distribution is positively skewed, negatively skewed, and symmetric. The study using kurtosis and skewness is helpful to determine the suitable distribution for the data. Based on the information is obtained from these tests, the decision-makers can decide which distribution is suitable for the analysis of the data. The classical measures of kurtosis and skewness are calculated based on the third and fourth moments of mean which cause the increase in bias for the non-normal distribution case. Reference [3] introduced the alternative forms of measures of kurtosis and skewness based on robust estimators. References [4] and [5] pointed out that conventional estimators are affected by outliers. [6] worked on a data trimming approach.

References [7] and [8] pointed that the measurements are imprecise and in-intervals in the variety of fields. In the case of the imprecise data, the traditional measures of kurtosis and skewness cannot be applied. In such situations, the statistical tests using fuzzy logic are applied. References [9], [10] and [11] applied the statistical tests under the fuzzy approach. Reference [12] worked on non-parametric test for the interval data.

The fuzzy logic does not provide information about the measure of indeterminacy. The neutrosophic logic has an edge over fuzzy logic as it gives information about the measures of truth, falseness, and indeterminacy, see [13]. According to [13], neutrosophic logic is more efficient than fuzzy logic and interval-based analysis. The applications of the neutrosophic logic can be seen in References [14 –25] and [26]. [27] introduced the idea of neutrosophic statistics which is applied when the data is uncertain, vague, and indeterminate. Neutrosophic statistics is a generalization of classical statistics and gives additional information about the measure of indeterminacy. References [28] and [29] presented the methods to analyze the neutrosophic numbers. Reference [30] proposed a new Mann-Whitney test under indeterminacy. Some applications of tests using the neutrosophic statistics can be seen in [31] and [32].

The tests of kurtosis and skewness under neutrosophic statistics can be more efficient than the tests under classical statistics and tests under fuzzy logic. By exploring the literature and best of our knowledge, there is no work on tests of kurtosis and skewness under an indeterminate environment. In this paper, we will present the estimators of kurtosis and skewness under neutrosophic statistics. The application of the proposed tests will be given with the data associated with heart disease. It is expected that the proposed tests will be helpful to see the shape of the medical data in the presence of uncertainty.

2 The proposed tests

In the following sections, we will present the tests of kurtosis and skewness under the neutrosophic statistics. We will first introduce the two estimators of kurtosis under the neutrosophic statistics and then the estimator of skewness under the neutrosophic statistics, respectively. Suppose that a_Ni and b_Ni are two neutrosophic numbers of size n_N, where i = 1, 2, 3, …, n_N. Let X_Ni = a_Ni + b_NiI_N ; I_Nε [I_L, I_U] be a neutrosophic random number where a_Ni and b_NiI_N are the determined part and indeterminate part of neutrosophic numbers, respectively. Note here that I_Nε [I_L, I_U] is an indeterminate interval. The neutrosophic number X_Niε [X_Li, X_Ui] reduces to the classical random number if X_Li = 0. Based on this information, the neutrosophic average can be written as ${\bar{X}}_{N} = {\bar{a}}_{N} + {\bar{b}}_{N} I_{N}; I_{N} ε [I_{L}, I_{U}]$ (1) where ${\bar{a}}_{N} = \frac{1}{n_{N}} \sum_{i = 1}^{n_{N}} a_{i}$ and ${\bar{b}}_{N} = \frac{1}{n_{N}} \sum_{i = 1}^{n_{N}} b_{i}$ .

2.1 The first estimator of kurtosis

References [33] and [34] argued that the classical kurtosis estimator is very sensitive to detect the outlier in the data. Reference [34] suggested an alternative kurtosis estimator see the nature of the shape of the data. We now extend [34] estimator of kurtosis, say K_Nr2 under the neutrosophic statistics. Suppose that X_N(i) = a_N(i) + b_N(i)I_N ; I_Nε [I_L, I_U] be an ordered neutrosophic number. Let U_N0.05 denote the mean of upper 5% of X_N(i), L_N0.05 is the mean of lower 5% of X_N(i), U_N0.5 and L_N0.5 are lower and upper 50% of X_N(i).

The values of U_N0.05ε [U_L0.05, U_U0.05] can be computed by $U_{N 0.05} = {\bar{a}}_{N 0.05} + {\bar{b}}_{N 0.05} I_{N 0.05}; I_{N 0.05} ε [I_{L 0.05}, I_{U 0.05}]$ (2)

The values of L_N0.05ε [L_L0.05, L_U0.05] can be computed by $L_{N 0.05} = {\bar{a}}_{N 0.05} + {\bar{b}}_{N 0.05} I_{N 0.05}; I_{N 0.05} ε [I_{L 0.05}, I_{U 0.05}]$ (3)

The values of U_N0.5ε [U_L0.5, U_U0.5] can be computed by $U_{N 0.5} = {\bar{a}}_{N 0.5} + {\bar{b}}_{N 0.05} I_{N 0.5}; I_{N 0.5} ε [I_{L 0.5}, I_{U 0.5}]$ (4)

The values of L_N0.5ε [L_L0.5, L_U0.5] can be computed by $L_{N 0.5} = {\bar{a}}_{N 0.5} + {\bar{b}}_{N 0.05} I_{N 0.5}; I_{N 0.5} ε [I_{L 0.5}, I_{U 0.5}]$ (5)

Using the above-mentioned equations, the estimator K_Nr2ε [K_Lr2, K_Ur2] is defined as follows $K_{Nr} 2 = \frac{U_{N 0.05} - L_{N 0.05}}{U_{N 0.5} - L_{N 0.5}}; K_{Nr} 2 ε [K_{Lr} 2, K_{Ur} 2]$ (6)

According to [3] and [35], the data has light-tailed distribution if K_Nr2 values are less than 2, the data has the medium tailed normal distribution if the values of K_Nr2 from 2 to 2.6, the data has heavily-tailed distribution such as double-exponential distribution if the values of K_Nr2 from 2.6 to 3.2 and the data follows a heavy-tailed distribution such as Cauchy distribution if K_Nr2 values are larger than 3.2.

2.2 The second estimator of kurtosis

[3] provided another estimator of kurtosis say K_Nr3 to study the nature of the data. We now present the modification of the estimator K_Nr3 under neutrosophic statistics. Let U_N0.20 denote the mean of upper 20% of X_N(i), L_N0.20 is the mean of the lower 20% of X_N(i). Based on this information, the estimator of kurtosis K_Nr3ε [K_Lr3, K_Ur3] is defined by $K_{Nr} 3 = \frac{U_{N 0.20} - L_{N 0.20}}{U_{N 0.5} - L_{N 0.5}}; K_{Nr} 2 ε [K_{Lr} 2, K_{Ur} 2]$ (7)

According to [36], the numerator and denominator of K_Nr3 is an excellent estimate of standard deviation (SD) of normal distribution and double exponential distribution, respectively. [3] and [35] pointed out that the data has light-tailed distribution if K_Nr3 is less than 1.81, the distribution is said to be medium-tailed distribution if K_Nr3 from 1.81 to 1.87 and the data has follows heavy-tailed distribution if K_Nr3 is larger than 1.87.

2.3 The estimator of skewness

The estimator of skewness is used to determine either the distribution is positively (+vely) skewed, negatively (-vely) skewed and normal. [3] proposed this estimator is based on the trimmed values of the order data. Let m_N0.25 denote the mean of upper and lower values trimmed by 25%. The estimator of skewness under neutrosophic statistics say S_Nr2ε [S_Lr2, S_Ur2] is defined by $S_{Nr} 2 = \frac{U_{N 0.05} - m_{N 0.25}}{m_{N 0.25} - L_{N 0.05}}; S_{Nr} 2 ε [S_{Lr} 2, S_{Ur} 2]$ (8)

As mentioned by [37] and [38], the distribution is said to be symmetrical if the values of S_Nr2 from 0.5 to 2, the distribution is said to be –vely skewed if values of S_Nr2 is less than 0.5 and the distribution is said to be+vely skewed distribution if S_Nr2 is larger than 2.

3 Applications of the proposed tests on heart disease data

In this section, we will discuss the application of the proposed tests using medical data about heart disease. The data consists of three variables namely pulse rate (PR), systolic pressure (SP), and diastolic pressure (DP). Reference [39] provided some basic analysis of the data. The data of 60 patients are reported in Table 1 where the first 11 values are the original data and selected from [39] and the remaining observations are generated through simulation. The cardiologist is interested to see the nature of the three variables associated with heart disease. From Table 1, it can be noted that the three variables are measured in intervals rather than the exact values. Therefore, the existing tests of kurtosis and skewness under classical statistics cannot be applied to study the nature of these variables. Therefore, the proposed tests of kurtosis and skewness can be applied to see either the data of three variables follow the normal distribution or not.

Table 1
The medical data

Patient # Pulse rate Systolic pressure Diastolic pressure

a _i b _i a _i b _i a _i b _i

1 44 68 90 100 50 70

2 62 72 90 130 70 90

3 56 90 140 180 90 100

4 70 112 110 142 80 108

5 54 72 90 100 50 70

6 70 100 130 160 80 110

7 63 75 60 100 140 150

8 72 100 130 160 76 90

9 76 98 110 190 70 110

10 86 96 138 188 90 110

11 86 100 110 150 78 100

12 45 90 87 127 60 118

13 70 85 100 112 79 96

14 63 80 104 138 84 133

15 70 81 122 134 110 127

16 93 95 89 125 64 114

17 41 87 102 139 82 134

18 44 84 105 126 86 116

19 54 72 81 90 52 65

20 41 53 92 132 67 125

21 65 83 94 140 70 136

22 44 94 99 117 77 103

23 40 76 114 145 99 143

24 45 92 98 121 76 108

25 73 98 119 139 105 135

26 47 95 87 99 61 78

27 68 76 90 103 64 82

28 57 73 116 147 136 146

29 69 97 94 141 70 137

30 55 76 87 113 60 97

31 44 51 87 122 60 110

32 48 66 133 143 126 140

33 73 91 122 145 110 142

34 54 87 103 139 117 135

35 74 94 83 107 55 88

36 60 80 101 110 80 93

37 41 90 81 94 52 70

38 77 94 106 132 87 124

39 40 94 95 99 72 78

40 45 84 121 141 108 138

41 76 83 112 144 97 142

42 41 98 91 127 65 117

43 50 74 89 109 63 92

44 53 72 84 148 56 147

45 48 75 83 125 55 114

46 55 97 130 139 121 134

47 80 97 80 123 50 112

48 45 86 101 122 81 111

49 60 94 105 132 86 124

50 63 86 107 146 89 145

51 61 90 97 146 75 145

52 40 98 86 121 59 109

53 68 71 87 105 60 86

54 66 91 141 143 137 140

55 50 67 81 105 51 86

56 52 95 129 141 120 137

57 47 96 84 106 56 88

58 75 87 88 117 62 103

59 58 73 86 120 58 107

60 51 89 93 148 69 147

Patient #	Pulse rate	Systolic pressure	Diastolic pressure
1	44	68	90	100	50	70
2	62	72	90	130	70	90
3	56	90	140	180	90	100
4	70	112	110	142	80	108
5	54	72	90	100	50	70
6	70	100	130	160	80	110
7	63	75	60	100	140	150
8	72	100	130	160	76	90
9	76	98	110	190	70	110
10	86	96	138	188	90	110
11	86	100	110	150	78	100
12	45	90	87	127	60	118
13	70	85	100	112	79	96
14	63	80	104	138	84	133
15	70	81	122	134	110	127
16	93	95	89	125	64	114
17	41	87	102	139	82	134
18	44	84	105	126	86	116
19	54	72	81	90	52	65
20	41	53	92	132	67	125
21	65	83	94	140	70	136
22	44	94	99	117	77	103
23	40	76	114	145	99	143
24	45	92	98	121	76	108
25	73	98	119	139	105	135
26	47	95	87	99	61	78
27	68	76	90	103	64	82
28	57	73	116	147	136	146
29	69	97	94	141	70	137
30	55	76	87	113	60	97
31	44	51	87	122	60	110
32	48	66	133	143	126	140
33	73	91	122	145	110	142
34	54	87	103	139	117	135
35	74	94	83	107	55	88
36	60	80	101	110	80	93
37	41	90	81	94	52	70
38	77	94	106	132	87	124
39	40	94	95	99	72	78
40	45	84	121	141	108	138
41	76	83	112	144	97	142
42	41	98	91	127	65	117
43	50	74	89	109	63	92
44	53	72	84	148	56	147
45	48	75	83	125	55	114
46	55	97	130	139	121	134
47	80	97	80	123	50	112
48	45	86	101	122	81	111
49	60	94	105	132	86	124
50	63	86	107	146	89	145
51	61	90	97	146	75	145
52	40	98	86	121	59	109
53	68	71	87	105	60	86
54	66	91	141	143	137	140
55	50	67	81	105	51	86
56	52	95	129	141	120	137
57	47	96	84	106	56	88
58	75	87	88	117	62	103
59	58	73	86	120	58	107
60	51	89	93	148	69	147

The necessary neutrosophic descriptive of three variables for estimator K_Nr2 are shown in Table 2. The necessary neutrosophic descriptive of three variables for estimator K_Nr3 are shown in Table 3. The necessary neutrosophic descriptive of three variables for estimator S_Nr2 are shown in Table 4. The values of the three estimators K_Nr2, K_Nr3 and S_Nr2 are shown in Table 5. From Table 5, it can be seen that according to the estimator K_Nr2, the PR follows the medium-tailed neutrosophic normal distribution. It is interesting to note that the determined part of the variable SP follows the medium-tailed neutrosophic normal distribution and the indeterminate part follows the neutrosophic heavy-tailed distribution such as neutrosophic double exponential distribution. The variable DP follows the medium-tailed neutrosophic normal distribution. The estimator K_Nr3 indicates that PR, SP, and DP follow the light-tailed distributions. The estimator S_Nr2 indicate that the variables PR and SP have a symmetric distribution. For the variable DP, the determinate part follows the –vely neutrosophic skewed distribution (a neutrosophic distribution having tail lengthy on the left side), and the indeterminate part follows the neutrosophic symmetric distribution. From this study, it can be concluded that the proposed tests can guide the cardiologist in decision-making.

Table 2

Estimator of Kurtosis K_Nr2

Variables	U _N0.05	L _N0.05	U _N0.5	L _N0.5	K_Nr2
Pulse Rate	[40.0,56.7]	[88.3,104.0]	[47.1,75.6]	[70.1,95.1]	[2.1,2.4]
Systolic pressure	[73.7,94.3]	[139.7,186.0]	[86.9,112.5]	[115.3,147.1]	[2.3,2.65]
Diastolic pressure	[50.0,68.3]	[137.7,148.0]	[60.9,93.6]	[97.6,133.2]	[2.4,2.01]

Table 3

Estimator of Kurtosis K_Nr3

Variables	U _N0.2	L _N0.2	U _N0.5	L _N0.5	K_Nr3
Pulse Rate	[42.1,67.5]	[78.4,99.3]	[47.1,75.6]	[70.1,95.1]	[1.6,1.63]
Systolic pressure	[81.3,100.7]	[129.6,159]	[86.9,112.5]	[115.3,147.1]	[1.7,1.68]
Diastolic pressure	[53.7,79.3]	[119.08,143.8]	[60.9,93.6]	[97.6,133.2]	[1.78,1.62]

Table 4

Estimator of Skewness S_Nr2

Variables	U _N0.05	m _N0.25	m _N0.25	L _N0.05	S_Nr2
Pulse Rate	[40.0,56.7]	[57.6,86.9]	[57.6,86.9]	[88.3,104.0]	[0.57,1.76]
Systolic pressure	[73.7,94.3]	[97.6,130.2]	[97.6,130.2]	[139.7,186.0]	[0.56,0.64]
Diastolic pressure	[50.0,68.3]	[74.1,114.8]	[74.1,114.8]	[137.7,148.0]	[0.37,1.40]

Table 5

Nature of the distributions

Variables	K_Nr2	Status	K_Nr3	Status	S_Nr2	Status
Pulse Rate	[2.1,2.4]	Normal	[1.6,1.63]	Light-tailed	[0.57,1.76]	Symmetric
Systolic pressure	[2.3,2.65]	Normal &heavy tailed	[1.7,1.68]	Light-tailed	[0.56,0.64]	Symmetric
Diastolic pressure	[2.4,2.01]	Normal	[1.78,1.62]	Light-tailed	[0.37,1.40]	-vely skewed &Symmetric

4 Comparative study based on heart disease data

In this section, we will compare the efficiency of the proposed tests over the existing tests under classical statistics proposed by [34]. We will show that the proposed tests are efficient in measure of indeterminacy. The proposed tests reduce to the tests by [34] if no indeterminacy is recorded in the data. For a fair comparison, we will consider the same data was given in the last section. The neutrosophic forms of three variables PR, SP, and DP are shown in Table 6. Note here that the first part of neutrosophic forms presents the determined part of the tests which is the same as in [34] and the second part of neutrosophic forms presents the indeterminate part. For example, for variable DP, the neutrosophic form for S_Nr2 is S_Nr2 = 0.37 + 1.40I_N ; I_N ∈ [0, 0.74]. According to the theory of neutrosophic, the proposed tests reduce to tests under classical statistics when I_L = 0. Therefore, the value 0.37 presents the value of skewness tests under classical statistics and the second value 1.40I_N show the value of the indeterminate part. From Table 6, it can be seen that the proposed test gives information about the measure of indeterminacy which the existing tests do not provide. For the above-mentioned neutrosophic form, the measure of indeterminacy is 0.74. From this study, it can be concluded that the proposed tests provide the results in the indeterminate intervals with the measure of indeterminacy. On the other hand, the existing tests give only the exact values of the tests which may mislead the cardiologist in making the decision about the nature of variables associated with heart disease. The proposed test is also a generalization of the test based on fuzzy logic and interval-based analysis. For the neutrosophic form S_Nr2 = 0.37 + 1.40I_N ; I_N ∈ [0, 0.74], S_Nr2 is also better structured, since we know that 0.37 is determined part, and 1.40I_N is the fluctuating part around 0.37. In addition, I_N may not necessarily be an interval, but any subset that is not the case of fuzzy logic and interval-based analysis. Therefore, the proposed test using neutrosophic statistics is more efficient than fuzzy logic and interval-based analysis.

Table 6
Neutrosophic form of three variables

Variables K_Nr2 K_Nr3 S_Nr2

PR 2.1 + 2.4I_N ; I_N ∈ [0, 0.125] 1.6 + 1.63I_N ; I_N ∈ [0, 0.018] 0.57 + 1.76I_N ; I_N ∈ [0, 0.68]

SP 2.3 + 2.65I_N ; I_N ∈ [0, 0.132] 1.7–1.68I_N ; I_N ∈ [0, 0.11] 0.56 + 0.64I_N ; I_N ∈ [0, 0.125]

DP 2.4–2.01I_N ; I_N ∈ [0, 0.20] 1.78–1.62I_N ; I_N ∈ [0, 0.099] 0.37 + 1.40I_N ; I_N [0, 0.74]

Variables	K_Nr2	K_Nr3	S_Nr2
PR	2.1 + 2.4I_N ; I_N ∈ [0, 0.125]	1.6 + 1.63I_N ; I_N ∈ [0, 0.018]	0.57 + 1.76I_N ; I_N ∈ [0, 0.68]
SP	2.3 + 2.65I_N ; I_N ∈ [0, 0.132]	1.7–1.68I_N ; I_N ∈ [0, 0.11]	0.56 + 0.64I_N ; I_N ∈ [0, 0.125]
DP	2.4–2.01I_N ; I_N ∈ [0, 0.20]	1.78–1.62I_N ; I_N ∈ [0, 0.099]	0.37 + 1.40I_N ; I_N [0, 0.74]

5 Simulation study

To see the effect of measure of the indeterminacy of the three estimators under neutrosophic statistics, a simulation study was performed for variable DP. To see the trends in the three estimators, various values of the upper value of indeterminacy I_U are considered. The values of estimators K_Nr2, K_Nr3 and S_Nr2 are shown in Table 7.

Table 7
The effect of indeterminacy on estimators

I _U K_Nr2 Status K_Nr3 Status S_Nr2 Status

0 0.37 light-tailed distribution 2.4 heavy-tailed distribution 1.78 Symmetrical distribution

0.1 0.51 light-tailed distribution 2.199 heavy-tailed distribution 1.618 Symmetrical distribution

0.2 0.65 light-tailed distribution 1.998 heavy-tailed distribution 1.456 Symmetrical distribution

0.3 0.79 light-tailed distribution 1.797 light-tailed distribution 1.294 Symmetrical distribution

0.4 0.93 light-tailed distribution 1.596 light-tailed distribution 1.132 Symmetrical distribution

0.5 1.07 light-tailed distribution 1.395 light-tailed distribution 0.97 Symmetrical distribution

0.6 1.21 light-tailed distribution 1.194 light-tailed distribution 0.808 Symmetrical distribution

0.7 1.35 light-tailed distribution 0.993 light-tailed distribution 0.646 Symmetrical distribution

0.74 1.406 light-tailed distribution 0.9126 light-tailed distribution 0.5812 Symmetrical distribution

0.8 1.49 light-tailed distribution 0.792 light-tailed distribution 0.484 –vely skewed distribution

0.9 1.63 light-tailed distribution 0.591 light-tailed distribution 0.322 –vely skewed distribution

1 1.77 light-tailed distribution 0.39 light-tailed distribution 0.16 –vely skewed distribution

I _U	K_Nr2	Status	K_Nr3	Status	S_Nr2	Status
0	0.37	light-tailed distribution	2.4	heavy-tailed distribution	1.78	Symmetrical distribution
0.1	0.51	light-tailed distribution	2.199	heavy-tailed distribution	1.618	Symmetrical distribution
0.2	0.65	light-tailed distribution	1.998	heavy-tailed distribution	1.456	Symmetrical distribution
0.3	0.79	light-tailed distribution	1.797	light-tailed distribution	1.294	Symmetrical distribution
0.4	0.93	light-tailed distribution	1.596	light-tailed distribution	1.132	Symmetrical distribution
0.5	1.07	light-tailed distribution	1.395	light-tailed distribution	0.97	Symmetrical distribution
0.6	1.21	light-tailed distribution	1.194	light-tailed distribution	0.808	Symmetrical distribution
0.7	1.35	light-tailed distribution	0.993	light-tailed distribution	0.646	Symmetrical distribution
0.74	1.406	light-tailed distribution	0.9126	light-tailed distribution	0.5812	Symmetrical distribution
0.8	1.49	light-tailed distribution	0.792	light-tailed distribution	0.484	–vely skewed distribution
0.9	1.63	light-tailed distribution	0.591	light-tailed distribution	0.322	–vely skewed distribution
1	1.77	light-tailed distribution	0.39	light-tailed distribution	0.16	–vely skewed distribution

From Table 7, it can be seen that the values of estimator K_Nr2 increases from 0.37 to 1.77 when I_U changes from 0 to 1. As K_Nr2 values at all values of I_U are less than 2, therefore, the nature of distribution is light-tailed distribution. From the analysis, it can be concluded that I_U does not effect K_Nr2. The nature of distribution based on the values of K_Nr3 is different when I_U ⩽ 0.2. For I_U ⩽ 0.2, the values of K_Nr3 are larger than 1.87 which indicate the nature of the distribution is heavy-tailed distribution. On the other hand, I_U > 0.2, the values of K_Nr3 <1.81 shows the nature of the distribution is light-tailed distribution. The values of S_Nr2 are from 0.5 to 2 when I_U ⩽ 0.74 which indicates that the nature of the distribution is symmetrical. The nature of data is –vely skewed distribution for the other values of I_U. Based on the simulation study, it can be concluded for K_Nr2 does not affect by I_U. On the other hand, I_U affects the estimators K_Nr3 and S_Nr2. The effects of I_U on other variables can be studied on the same lines.

6 Concluding remarks

In this paper, tests of skewness and kurtosis were introduced under neutrosophic statistics. The necessary measures and neutrosophic forms of these estimators were introduced. The application of the proposed tests was given using the data associated with heart diseases. From the real application, it can be seen that the proposed tests are the generalization of the existing tests. The proposed tests provide the results in indeterminacy intervals which are needed in the presence of uncertainty. The proposed test can be applied in medical science, decision science, big data analysis, and social science. The proposed test using big data can be extended as future research.

Footnotes

Acknowledgments

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of the paper.

Author contribution

M.A and M.A.B wrote the paper.

Data availability

The data is taken from [].

Funding statement

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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