# Correlated Proportions Test under Indeterminacy

Authors: Abdullah M. Almarashi; Muhammad Aslam
Journal: Journal of Mathematics
 (2021)

---

## Abstract

The existingZ-test for correlated proportions under classical statistics is applied when observations in the data are exact and precise. In the current paper, Z-test for correlated proportions is proposed to deal with the data having indeterminate observations. The statistic of Z-test for correlated proportions under neutrosophy is introduced in the paper. The testing procedure of the proposed test is given. The application of the proposed test is given using the price of oil data. From the application, it is clear that the proposed test outperforms the existing Z-test for correlated proportions under classical statistics in terms of flexibility, adequacy, and information. The simulation study also shows the effect of indeterminacy on the statistic of the proposed test.


---

## Body

## 1. Introduction

TheZ-test is a statistical test that is used to see the differences between two population variances are equal or not. This test is applied under the assumption that the data are obtained from the normal population and samples are independent. The Z-test for correlated proportions is applied to investigate the difference between two correlated proportions. The Z-test for correlated proportions is applied to test the null hypothesis that proportions of two populations are the same vs. the alternative hypothesis that there is a significant difference between two population proportions. The Z-test has the limitation that it is applied when the sample size is greater than thirty. The authors of [1–4] and [5] worked on the applications of statistical tests.The statistical tests under classical statistics may mislead if the observations in the data are indeterminate, imprecise, and fuzzy. Viertl [6] mentioned that “statistical data are frequently not precise numbers but more or less nonprecise, also called fuzzy. Measurements of continuous variables are always fuzzy to a certain degree.” The fuzzy-based statistical tests are considered quite adequate to deal with fuzzy data. The authors of [7–14] and [15] presented excellent work on statistical tests using fuzzy logic.The neutrosophic statistics as an extension of classical statistics was introduced in [16] using the idea of neutrosophic logic introduced in [17]. The efficiency of neutrosophic logic was discussed in [18]. The authors of [19–21] [22, 23] applied neutrosophic logic to deal with real problems. The authors of [24–29] and [30] used neutrosophic statistics to analyze uncertain, indeterminate, and interval data. More details on neutrosophic statistics can be seen in [31].The existingZ-test for correlated proportions under classical statistics cannot be applied for testing in the presence of uncertainty, indeterminacy, and neutrosophy in the data. The Z-test for correlated proportions under neutrosophic statistics should be introduced to analyze such a type of data. In this paper, the Z-test for correlated proportions under indeterminacy is introduced. The neutrosophic statistic for the proposed test is also introduced. The necessary steps to carry out the proposed test are given. The proposed test is analyzed using the oil payoff data. With the application and simulation studies, it is expected that the proposed test will be flexible, informative, and adequate in uncertainty.


## 2. Proposed Test

In case of uncertainty and indeterminacy in the observations, the existingZ-test for correlated proportions cannot be applied for testing the significant differences between two population proportions. In this section, Z-test for correlated proportions will be designed when the data are imprecise, indeterminate, and neutrosophic under the assumption that the neutrosophic sample nNεnL,nU is quite large. The general form of 2×2 contingency table having imprecise, indeterminate, and neutrosophic observations is introduced in Table 1.Table 1
2×2 contingency table 1st variable.
2nd variableColumn 1Column 2TotalRow 1aNεaL,aUbNεbL,bUaN+bN=N1NRow 2cNεcL,cUdNεdL,dUcN+dN=N2NTotalaN+cN=N0.1NbN+dN=N0.2NNNεNL,NUNote.NNεNL,NU denotes the overall neutrosophic total. The 2×2 contingency table in neutrosophic form is introduced in Table 2.Table 2
Neutrosophic form of2×2 contingency table 1st variable.
2nd variableColumn 1Column 2TotalRow 1aN=aL+aU;IaNεIaL,IaUbN=bL+bU;IbNεIbL,IbUN1N.=N1L.+N1U.;I1N.εI1L.,I1U.Row 2cN=cL+cU;IcNεIcL,IcUdN=dL+dU;IdNεIdL,IdUN2N.=N2L.+N2U.;I2N.εI2L.,I2U.TotalN0.1N=N0.1L+N0.1U;I0.1NεI0.1L,I0.1UN.2N=N.2L+N.2U;I.2NεI.2L,I.2UNN=NL+NUINεIL,IUIn Table2, the first part of the neutrosophic form presents the classical statistics (determined part) and the second part shows the indeterminate part with an associated measure of indeterminacy. To test the hypothesis that there is a difference between two population proportions, the neutrosophic statistic ZNεZL,ZU is defined as(1)ZN=bN−cNNNσN;bNεbL,bU,cNεcL,cU,σNεσL,σU,where σNεσL,σU is defined as(2)σN=bN+cN−bN−cN2/NNNNNN−1;bNεbL,bU,cNεcL,cU,NNεNL,NU.The neutrosophic form of statisticZNεZL,ZU is defined as(3)ZN=ZL+ZUIZN;IZNεIZL,IZU.The neutrosophic form ofZNεZL,ZU consists of two parts namely the determined part and indeterminate part. The first part ZL denotes the determined part and ZUIZN;IZNεIZL,IZU denotes the indeterminate part. Note that IZNεIZL,IZU is a measure of indeterminacy associated with the statistic ZNεZL,ZU. The proposed test statistic ZNεZL,ZU reduces to statistic under classical statistics when IZL=0.


## 3. Application Using Oil Price Data

The application of the proposed test is given using the data of crude oil prices. For the application, five states, stock A, and stock B are chosen. Two categories arew1: price is decreased by 1%-2% and w2: the price is increased by 1%-2%. The data are taken from [32] and reported in Table 3. The neutrosophic form of the payoff data is shown in Table 4. The decision maker is interested to investigate the difference between the proportions of prices in two stocks. From the fuzzy payoff matrix for stock A and stock B in Table 3, it can be seen that the payoff data is in the interval; therefore, the existing test under classical statistics cannot be applied to test the null hypothesis H0: the difference between the two correlated proportions is insignificant vs. the alternative hypothesis H1: the difference between the two correlated proportions is significant. The proposed test is quite adequate to be applied for testing H0 vs. H1 for the given data. The constant σNεσL,σU for the given data is calculated as(4)σN=32,35+24,25.5−32,35+24,25.52/98,10698,10698,106−1=0.0763,0.0732.Table 3
Payoff for two stocks.
w1w2TotalStock A[26, 27.5][32, 35][58, 62.5]Stock B[24, 25.5][16, 18][40, 43.5]Total[50, 53][48, 53][98, 106]Table 4
Neutrosophic form of payoff for two stocks.
w1w2TotalStock A26 + 27.5IN;INε0,0.05432 + 35IN;INε0,0.08558 + 62.5IN;INε0,0.065Stock B24 + 25.5IN;INε0,0.05916 + 18IN;INε0,0.140 + 43.5IN;INε0,0.069Total50 + 53IN;INε0,0.05648 + 53IN;INε0,0.09498 + 106IN;INε0,0.075The neutrosophic statisticZNεZL,ZU for the given data is calculated as(5)ZN=32,35−24,25.598,1060.0763,0.0732=0.0062,0.0065.The neutrosophic form of statisticZNεZL,ZU is given as ZN=0.0062+0.0065IZN;IZNε0,0.0506. The proposed test can be implemented in the following steps.Step 1 (stateH0): no difference between proportions of prices in two stocks vs. H1: difference exist between proportions of prices in two stocksStep 2: let the level of significanceα = 5% and the tabulated critical value be 1.96Step 3: we do not rejectH0 as ZNε0.0062,0.0065<1.96.The operational procedure of the proposed test for the real example is also shown in Figure1.Figure 1
The procedure of the proposed test for real example.


## 4. Comparative Study

The proposed test is a generalization of the existingZ-test for correlated proportions under classical statistics. The proposed test reduces to the classical test when no vague values are recorded in the data. The efficiency of the proposed Z-test for correlated proportions under neutrosophic statistics is compared with the existing Z-test for correlated proportions under classical statistics in terms of a measure of vagueness. The neutrosophic form of statistic ZNεZL,ZU is ZN=0.0062+0.0065IZN;IZNε0,0.0506, where the first value 0.0062 presents the value of statistic under classical statistics and the second part 0.0065IZN shows the indeterminate part. Note that IZNε0,0.0506 is the measure of vagueness associated with the statistic ZNεZL,ZU. From the neutrosophic form, it is clear that the proposed statistic ZNεZL,ZU has a lower value, that is, 0.0062 and an upper value, that is, 0.0065. The proposed test statistic ZNεZL,ZU is expressed in indeterminate interval rather than the exact value. In testing of hypothesis, when the data are given in interval, the decision makers can expect the value from 0.0062 to 0.0065. On the contrary, the existing test gives only a single (exact) value which is not adequate in uncertainty. From this comparison, it can be concluded that the proposed test is elastic than the existing Z-test for correlated proportions. Now, the comparison of the proposed test will be given in terms of a measure of indeterminacy. For testing the hypothesis when the probability of committing a type-I error is 0.05, the proposed test is interpreted as follows: the probability of accepting H0 is 0.95, the probability of committing an error is 0.05, and the measure of vagueness is 0.05. From the analysis, it is clear that the proposed test gives additional information about the measure of vagueness. Based on the analysis, it can be concluded that the proposed test is adequate, informative, and elastic than the existing Z-test for correlated proportions.


## 5. Simulation Study

In this section, the simulation study is carried to see the effect of indeterminacy on the statisticZNεZL,ZU. The various values of IZN are taken to see the effect of indeterminacy on the statistic ZNεZL,ZU. The values of statistic ZNεZL,ZU for various IZN are shown in Table 5. From Table 5, it can be seen that ZN = 0.0062 when IZN = 0. It is interesting note that, as the values of IZN is increased from 0 to 1, the values of ZU is also increased. It means that the indeterminacy parameter affects the values of ZNεZL,ZU. From this study, it is concluded that the decision makers should be careful in applying the test under uncertainty.Table 5
Simulation results.
IZNZN=0.0062+0.0065IZN[0, 0][0.0062, 0.0062][0, 0.05][0.0062, 0.006525][0, 0.1][0.0062, 0.00685][0, 0.15][0.0062, 0.007175][0, 0.2][0.0062, 0.0075][0, 0.25][0.0062, 0.007825][0, 0.3][0.0062, 0.00815][0, 0.35][0.0062, 0.008475][0, 0.4][0.0062, 0.0088][0, 0.45][0.0062, 0.009125][0, 0.5][0.0062, 0.00945][0, 0.55][0.0062, 0.009775][0, 0.6][0.0062, 0.0101][0, 0.7][0.0062, 0.01075][0, 0.8][0.0062, 0.0114][0, 0.9][0.0062, 0.01205][0, 1][0.0062, 0.0127]


## 6. Concluding Remarks

The existingZ-test for correlated proportions under classical statistics is applied when observations in the data are exact and precise. In the current paper, Z-test for correlated proportions was proposed to deal with the data having indeterminate observations. The statistic of Z-test for correlated proportions under neutrosophy was introduced in the paper. The proposed test was a generalization of the exiting Z-test for correlated proportions and applied when the data are in intervals. The real example and the comparative study show that the proposed test is efficient than the existing Z-test for correlated proportions in terms of flexibility, adequacy, and information. The proposed test for big data can be studied as future research.



---

*Source: 6564006-2021-10-22.xml*