Statistical Analysis for Food Quality in the Presence of Vague Information

(e present study introduces the neutrosophic statistical test to see investigate the difference between variances of two populations when correlation exits in pair observations. (e procedure and statistic of the proposed test under neutrosophic statistics are introduced in the paper. (e application of the proposed test is given using the food industry data. (e efficiency of the proposed test is compared with that of the existing test in terms of the measure of indeterminacy, flexibility, and information. From the real application and comparative studies, it is concluded that the proposed test is quite reasonable to apply in uncertainty.


Introduction
e F-test is applied for testing the equality of two population variances under the assumption that the data are obtained from the normal distribution. Usually, the F-tests are applied under the assumption that the data are independent and obtained from the normal distribution [1]; we also discussed the application of the F-test when the pair data are correlated. is F-test is applied to investigate the significant difference between the equality of two population variances with correlated pair data [2][3][4]. More applications about the statistical tests can be seen in [4][5][6][7][8] where the applications of statistical tests in various practical fields are provided.
Classical statistics-based tests cannot be applied to the observations in the data which are fuzzy, in intervals, and uncertain. e fuzzy-based tests are the alternative of classical statistical tests in these situations. As mentioned in [9], "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." e authors of [10][11][12][13][14][15][16][17][18][19][20][21][22] worked on various types of the statistical tests using fuzzy logic. e statistical tests based on fuzzy logic do not give information about the measure of indeterminacy. To overcome this shortcoming [23], we introduced neutrosophic logic as an extension of fuzzy logic. e efficiency of the neutrosophic logic over fuzzy analysis and intervalbased analysis was discussed in [24]. Recently, several applications of the neutrosophic logic were discussed in [25][26][27][28][29][30] and neutrosophic statistics were introduced as an extension of classical statistics. For more details on the neutrosophic statistics, see https://archive.org/details/ neutrosophic-statistics?tab�about, https://archive.org/ details/neutrosophic-statistics?tab�collection and http://fs. unm.edu/NS/NeutrosophicStatistics.htm. e efficiency of neutrosophic statistics over classical statistics was proven in [31][32][33][34][35][36][37].
e existing F-test for testing the equality of variances for correlated data under classical statistics cannot be applied in the presence of imprecise data. By exploring the literature and to the best of our knowledge, we did not find any work on the F-test for testing the equality of variances for correlated data under neutrosophic statistics. In this paper, the work when observations are correlated under neutrosophic statistics will be presented. e operational procedure of the proposed test will be given for testing the hypothesis of equality of two variances. e application of the proposed test will be given in the data taken from the food industry. It is expected that the proposed test will be efficient than the existing test in terms of information, flexibility, and adequacy.

The Proposed F-Test for Variances with Correlated Data
As mentioned before, the existing F-test for two population variances when pair data are correlated can be applied only when all observations are determined, certain, and exact. In this section, the F-test for two population variances when pair data are correlated will be introduced when the data is an interval, indeterminate, and neutrosophic. e main objective of the proposed test is to investigate the difference between two pupation variances when the data are paired and have a correlation. In addition, it is assumed that the data follow the neutrosophic normal distribution. Let σ 2 1N and σ 2 2N be the neutrosophic variances of the first and the second population, respectively. e proposed test will be applied for testing the null hypothesis. e proposed test will be applied for testing the null hypothesis H 0 : } is a pair of neutrosophic observations of neutrosophic sample size n N ∈ [n L , n U ]. Note that X 1L , . . . , X nL and Y 1L , . . . , Y nL are the determined part of pair observations and X 1U I 1N , . . . , X nU I nXN and Y 1U I 1N , . . . , Y nU I nYN are the indeterminate part of the same pair observations. Note also that I nXN ∈ [I nXL , I nXU ] and I nYN ∈ [I nYL , I nYU ] are measures of indeterminacy associated with the neutrosophic pair observations. Based on the information and by following the work in [35,36], the neutrosophic means are defined as (1) e neutrosophic variances are defined as Under e neutrosophic correlation defined by the work in [34] is given by ; n N ∈ n L , n U .
2 Journal of Food Quality In the neutrosophic form of c N F N ∈ [c L F U , c L F U ], the first part c L F L presents the determined part and c U F U I c N F N is the indeterminate part. e proposed form of quotient reduces to classical statistics when I c L F L � 0. e application of the proposed test is discussed using Figure 1.

Application Using Food Data
e application of the proposed test will be given on the data collected from the food industry. To keep the quality of the food, the food inspectors test the food for different characteristics such as taste, shape, and hardness. Similar examples were discussed in [22,38]. e evaluation of food by the inspector for product A and product B is shown in Table 1. From Table 1, it is clear that experts provide the food evaluation in indeterminate interval reporting the minimum value and the maximum value. e evaluation of food is imprecise data rather than the exact; therefore, the existing test is under classical statistics. For the data, the decision makers are interested to see whether the variances of both products have the same variances or not.
e neutrosophic form of c N F N ∈ [c L F U , c L F U ] is given as In the neutrosophic form of c N F N ∈ [c L F U , c L F U ], the first part 0.06 presents the determined part and 0.28I c N F N is the indeterminate part, where I c N F N ∈ [0, 0.7857] is the measure of indeterminacy associated with c N F N ∈ [c L F U , c L F U ]. e proposed form of quotient reduces to classical statistics when I c L F L � 0. e proposed test is implemented in the following steps: Step-1: state H 0 : Step-2: set the level of significance α � 0.05 Step-3: calculate c N F N ∈ [0.06, 0.28], and compared with the critical value from [1], it is 0.632 Step Reject H 0 : Figure 1: e operational process of the proposed test.

Journal of Food Quality
From the study, it is concluded that the neutrosophic variances of both food experts are the same. erefore, there is no significant difference between the variances.

Comparative Study Based on Food Data
It is noted that the proposed test is a generalization of the existing test under neutrosophic statistics. It is also worth noting that the proposed test reduces to the existing test under classical statistics when no indeterminacy is found in the data. erefore, the efficiency of the proposed test will be given in terms of the measure of indeterminacy, information, and flexibility. e neutrosophic form of

Concluding Remarks
e present study introduced the neutrosophic statistical test to investigate the difference between variances of two populations when the correlation was exsiting in pair observations. e procedure and statistic of the proposed test under neutrosophic statistics were introduced in this paper. e proposed test is an extension of the existing F-test when correlation exists in pair observations. e application of the proposed test was given using the food industry data. e comparative study showed the efficiency of the proposed test over the existing test in terms of knowledge, flexibility, and adequacy. e proposed test using big data can be considered as future research. e proposed test using various sampling schemes can also be considered for future research.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.