Neutrosophic F-Test for Two Counts of Data from the Poisson Distribution with Application in Climatology

Abstract

This paper addresses the modification of the F-test for count data following the Poisson distribution. The F-test when the count data are expressed in intervals is considered in this paper. The proposed F-test is evaluated using real data from climatology. The comparative study showed the efficiency of the F-test for count data under neutrosophic statistics over the F-test for count data under classical statistics.

1. Introduction

The F-test for two counts obtained from the Poisson distribution has been applied for the testing of the hypothesis that two counts differ significantly vs. the alternative hypothesis that there are two counts that do not differ significantly. The F-test for two counts is used to test whether the two counts exposed to different conditions (respectively) are significantly different (see Kanji [1]). The F-test works when the sample is obtained under the same conditions. This type of test is applied for testing the occurrence of counts in a fixed time and when the time interval is different. The details of the F-test when the counts are obtained from the Poisson distribution can be found in Kanji [1]. A powerful test for comparing the means of the Poisson distribution was presented by Krishnamoorthy and Thomson [2]. The application of the count data in the field of education was studied by Hilbe [3]. The application and goodness-of-fit test were studied by Puig and Weiß [4]. Various applications of this kind of test can be seen in [5,6,7,8,9,10].

Weather and wind forecasting and estimation are conducted using statistical models. The statistical tests are applied for testing the significance of the models used in the estimation and forecasting of wind speed. For the applications of statistical techniques and methods, the reader may refer to [11,12,13,14,15,16,17,18,19,20].

The F-test for two counts under classical statistics is usually applied when counts from the Poisson distribution are sure, certain, and determinate. Viertl [21] stated that “statistical data are frequently not precise numbers but more or less non-precise, also called fuzzy. Measurements of continuous variables are always fuzzy to a certain degree”. When the count data are recorded in an interval such as the minimum count and the maximum count, the F-test for two counts cannot be applied. In such a situation, the statistical test designed under fuzzy logic can be applied. The application of statistical tests under fuzzy logic can be seen in [22,23,24,25,26,27,28,29].

The idea of neutrosophic logic was introduced by Smarandache [30]. The efficiency of neutrosophic logic can be found in Smarandache [31]. Some applications of the neutrosophic logic were described in [32,33,34,35,36]. Neutrosophic statistics with applications were discussed by Smarandache [37]. The statistical methods to deal with neutrosophic numbers were discussed by Chen et al. [38,39]. Aslam [40] introduced the F-test to perform an analysis of the variance test. Various neutrosophic tests with applications can be seen in [41,42].

Generally, the F-test under classical statistics is applied for testing the equality of variance of two population variances. The existing F-test for two counts (Poisson distribution) is applied when counts are determined or all parameters involved are certain. The F-test under classical statistics cannot be utilized when counts are recorded in an interval. According to the literature, to the best of the author’s knowledge, no F-test for two counts (Poisson distribution) can be found when counts are recorded in the interval. The main focus of the paper was to develop the F-test under neutrosophic statistics when two counts are modeled from a Poisson distribution. Data from the weather department are applied for the illustration of the proposed F-test for counts. It is expected that the proposed F-test for counts is more efficient than the existing F-test for counts under classical statistics.

2. Methods

Kanji [1] reported the F-test when two counts follow a Poisson distribution. Kanji [1] presented the statistical test when counts are obtained in a fixed time and obtained over different time intervals. Mathematical proofs of the F-test for two counts when the number of events occurring in a fixed time and different periods of time are shown in the Appendix A (Theorems A1 and A2). Let us assume that $N_{1N}=N_{1L}+N_{1U}{I}_{1N};I_{1N}ϵ\left[I_{1L},I_{1U}\right]$ and $N_{2N}=N_{2L}+N_{2U}{I}_{2N};I_{2N}ϵ\left[I_{1L},I_{2U}\right]$ are two neutrosophic counts consisting of determinate and indeterminate parts. For the measure of $I_{1N}ϵ\left[I_{1L},I_{1U}\right]$ and $I_{2N}ϵ\left[I_{1L},I_{2U}\right]$ indeterminacy in counts, see [43,44,45,46,47,48,49].

Theorem 1.

The statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ in neutrosophic form when counts are obtained in the same period of time$\left(t_1=t_2\right)$ is given as

$$F_{N1}=F_{L1}+F_{U1}{I}_{F_{N1}};I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right].$$

Proof. 

The basic operation on neutrosophic numbers can be performed as described below.

Let $F_{N1}=F_{L1}+F_{U1}{I}_{F_N}$ and $F_{N2}=F_{L2}+F_{U2}{I}_{F_N}$ for $I_{F_N}ϵ\left[I_{F_L},I_{F_U}\right]$ be neutrosophic numbers obtained from the neutrosophic F-distribution (see [37]). According to [38], the following operation can be performed on the neutrosophic numbers:

$$F_{N1}+F_{N2}=\left(F_{L1}+F_{L2}\right)+\left(F_{U1}+F_{U2}\right)I_{F_N}=\left[F_{L1}+F_{L2}+F_{U1}{I}_{F_L}+F_{U2}{I}_{F_L},F_{L1}+F_{L2}+F_{U1}{I}_{F_U}+F_{U2}{I}_{F_U}\right],$$

or

$$F_{N1}+F_{N2}=\left[\left(F_{L1}+F_{L2}\right)+\left(F_{U1}+F_{U2}\right)I_{F_L},\left(F_{L1}+F_{L2}\right)+\left(F_{U1}+F_{U2}\right)I_{F_U}\right].$$

Suppose that $F_{L1}$ is the lower value, $F_{U1}{I}_{F_{N1}}$ is the upper value of the neutrosophic form, and $I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right]$ is the measure of indeterminacy.

According to [1,37,38] and Theorems A1 and A2, we have

$$F_{L1}=\frac{N_{1L}}{N_{2L}+1}\mathrm{and}{F}_{U1}=\frac{N_{1U}}{N_{2U}+1},$$

i.e., $F_{N1}ϵ\left[\frac{N_{1L}}{N_{2L}+1},\frac{N_{1U}}{N_{2U}+1}\right].$

Note that $N_{1L}$ and $N_{1U}$ are the lower and upper counts of the first population, respectively, and $N_{2L}$ and $N_{2U}$ are the lower and upper counts of the second population, respectively.

Using the basic operation, the neutrosophic form of $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ can be written as

$$F_{N1}=\frac{N_{1L}}{N_{2L}+1}+\frac{N_{1U}}{N_{2U}+1}{I}_{F_{N1}};I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right],$$

or

$$F_{N1}=F_{L1}+F_{U1}{I}_{F_{N1}};I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right],$$

where $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ presents the neutrosophic F-distribution $\left(2\left(N_{2N}+1\right),2N_{1N}\right)$ degree of freedom. □

Theorem 2.

To test the null hypothesis that$H_{0N}:{\mu }_{1N}={\mu }_{2N}$, the F-test, say $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$, is given as $F_{N1}=\frac{N_{1N}}{N_{2N}+1};F_{N1}ϵ\left[F_{L1},F_{U1}\right]$.

Proof. 

Let $N_{1N}ϵ\left[N_{1L},N_{1U}\right]$ and $N_{2N}ϵ\left[N_{2L},N_{2U}\right]$ be count in intervals for the first and the second populations, respectively. Let $N_{1N}=N_{1L}+N_{1U}{I}_{1N};I_{1N}ϵ\left[I_{1L},I_{1U}\right]$ and $N_{2N}=N_{2L}+N_{2U}{I}_{2N};I_{2N}ϵ\left[I_{1L},I_{2U}\right]$ be the neutrosophic forms of these counts, where $I_{1N}ϵ\left[I_{1L},I_{1U}\right]$ and $I_{2N}ϵ\left[I_{1L},I_{2U}\right]$ are the measures of indeterminacy for counts in population 1 and population 2, respectively. By following Theorems A1 and A2, we can proceed as described below.

From [1], the F-test for classical statistics is given by

$$F_1=\frac{N_1}{N_2+1}.$$

The F-test, say $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$, under neutrosophic statistics is given by

$$F_{N1}=\frac{\left[N_{1L},N_{1U}\right]}{\left[N_{2L},N_{2U}\right]+1};I_{1N}ϵ\left[I_{1L},I_{1U}\right],I_{2N}ϵ\left[I_{1L},I_{2U}\right].$$

The F-test under indeterminacy is expressed as

$$F_{N1}=\frac{N_{1L}+N_{1U}{I}_{1N}}{\left(N_{2L}+N_{2U}{I}_{2N}\right)+1};I_{1N}ϵ\left[I_{1L},I_{1U}\right],I_{2N}ϵ\left[I_{1L},I_{2U}\right].$$

□

Suppose that the first population mean is ${\mu }_{1N}$ and the second population mean is ${\mu }_{2N}$. To test the null hypothesis that $H_{0N}:{\mu }_{1N}={\mu }_{2N}$, the F-test, say $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$, based on neutrosophic counts $N_{1N}ϵ\left[N_{1L},N_{1U}\right]$ and $N_{2N}ϵ\left[N_{2L},N_{2U}\right]$ is defined as

$$F_{N1}=\frac{N_{1N}}{N_{2N}+1};F_{N1}ϵ\left[F_{L1},F_{U1}\right].$$

(1)

The statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ is modeled by a neutrosophic F-distribution with $\left(2\left(N_{2N}+1\right),2N_{1N}\right)$ degrees of freedom (see Aslam [50]). The statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ in neutrosophic form can be expressed as

$$F_{N1}=F_{L1}+F_{U1}{I}_{F_{N1}};I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right].$$

(2)

The proposed statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ is a generalization of the existing F-test for two counts of data. The statistic $F_{L1}$ presents the existing F-test for two counts of data. Note that $F_{U1}{I}_{F_{N1}};I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right]$ presents the indeterminate part, and $I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right]$ is a measure of uncertainty associated with $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$. The proposed statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ becomes the existing statistic when$I_{F_{L1}}=0$.

Theorem 3.

The proposed statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$in neutrosophic form over the different periods of time$t_1$and$t_2$is given as

$$F_{N2}=F_{L2}+F_{U2}{I}_{F_{N2}};I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right].$$

Proof. 

Let $t_1$ and$t_2$ be different times with counting rates $N_{1N}/t_1$ and $N_{2N}/t_2$, respectively. Suppose that $F_{L1}$ is the lower value, $F_{U1}{I}_{F_{N1}}$ is the upper value of the neutrosophic form, and $I_{F_{N1}}ϵ\left[I_{F_{L1}},I_{F_{U1}}\right]$ is the measure of indeterminacy.

According to [1,37] and Theorems 1 and 2, we have

$F_{L1}=\frac{\frac{1}{t_1}\left(N_{2L}+0.5\right)}{\frac{1}{ t_2}\left(N_{1L}+0.5\right)}$ and $F_{U1}=\frac{\frac{1}{t_1}\left(N_{2U}+0.5\right)}{\frac{1}{ t_2}\left(N_{1U}+0.5\right)},$

i.e., $F_{N1}ϵ\left[\frac{\frac{1}{t_1}\left(N_{2L}+0.5\right)}{\frac{1}{t_2}\left(N_{1L}+0.5\right)},\frac{\frac{1}{t_1}\left(N_{2U}+0.5\right)}{\frac{1}{t_2}\left(N_{1U}+0.5\right)}\right].$

Note that $N_{1L}$ and $N_{1U}$ are the lower and upper counts of the first population, respectively, and $N_{2L}$ and $N_{2U}$ are the lower and upper counts of the second population, respectively.

Using the basic operation, the neutrosophic form of $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$ can be written as

$$F_{N2}=\frac{\frac{1}{t_1}\left(N_{2L}+0.5\right)}{\frac{1}{ t_2}\left(N_{1L}+0.5\right)}+\frac{\frac{1}{t_1}\left(N_{2U}+0.5\right)}{\frac{1}{ t_2}\left(N_{1U}+0.5\right)}{I}_{F_{N2}};I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right],$$

or

$$F_{N2}=F_{L2}+F_{U2}{I}_{F_{N2}};I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right],$$

where $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$ presents the neutrosophic F-distribution with $\left(2\left(N_{2N}+1\right),2N_{1N}\right)$ degrees of freedom. □

Theorem 4.

The statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$over different periods of time$t_1$and$t_2$and counting rates$N_{1N}/t_1$and$N_{2N}/t_2$is given by

$$F_{N2}=\frac{\frac{1}{t_1}\left(N_{2N}+0.5\right)}{\frac{1}{ t_2}\left(N_{1N}+0.5\right)};F_{N2}ϵ\left[F_{L2},F_{U2}\right].$$

Proof. 

According to [1] and Theorems A1 and A2, the F-test for classical statistics is given by

$$F_2=\frac{\frac{1}{t_1}\left(N_2+0.5\right)}{\frac{1}{ t_2}\left(N_1+0.5\right)}.$$

The F-test, say $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$, under neutrosophic statistics is given by

$$F_{N1}=\frac{\frac{1}{t_1}\left\{\left[N_{2L},N_{2U}\right]+0.5\right\}}{\frac{1}{t_2}\left\{\left[N_{1L},N_{1U}\right]+0.5\right\}+1};I_{1N}ϵ\left[I_{1L},I_{1U}\right],I_{2N}ϵ\left[I_{1L},I_{2U}\right].$$

The F-test under indeterminacy is expressed as

$$F_{N1}=\frac{\frac{1}{t_1}\left\{N_{2L}+N_{2U}{I}_{2N}+0.5\right\}}{\frac{1}{t_2}\left\{N_{1L}+N_{1U}{I}_{1N}+0.5\right\}+1};I_{1N}ϵ\left[I_{1L},I_{1U}\right],I_{2N}ϵ\left[I_{1L},I_{2U}\right],$$

when two counts are observed over diverse periods of time ${\mathrm{t}}_1$ and${\mathrm{t}}_2$, with the counting rates $N_{1N}/t_1$ and $N_{2N}/t_2$. The proposed statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$ is defined as

$$F_{N2}=\frac{\frac{1}{t_1}\left(N_{2N}+0.5\right)}{\frac{1}{ t_2}\left(N_{1N}+0.5\right)};F_{N2}ϵ\left[F_{L2},F_{U2}\right].$$

(3)

The proposed statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$ in neutrosophic form can be expressed as

$$F_{N2}=F_{L2}+F_{U2}{I}_{F_{N2}};I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right].$$

(4)

□

The existing F-test is a special case of the proposed statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$. The statistic $F_{L2}$ presents the existing F-test for two counts of data. Note that $F_{U2}{I}_{F_{N2}};I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right]$ presents the indeterminate part and $I_{F_{N2}}ϵ\left[I_{F_{L2}},I_{F_{U2}}\right]$ is a measure of the uncertainty associated with $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$. The proposed statistic $F_{N2}ϵ\left[F_{L2},F_{U2}\right]$ becomes the existing statistic when $I_{F_{L2}}=0$.

3. Application

The US daily and monthly weather recorded data which broke records are used to show the application of the proposed F-test. The daily or monthly records are noted in intervals rather than the exact number. The record weather data follow a Poisson distribution as these are count data; the events are independent, the average rate of record broken data during the specified time can be calculated, and two records are assumed to be not noted in the same time period. The low minimum, low maximum, high minimum, and high maximum data for daily and monthly records were accessed on 7 January 2021, through https://www.ncdc.noaa.gov/cdo-web/datatools/records. The daily record and monthly record data are presented in

Table 1. US daily records.

Period	Low Min, Low Max	High Min, High Max	${\mathit{F}}_{\mathit{N}1}\mathit{ϵ}\left[{\mathit{F}}_{\mathit{L}1},{\mathit{F}}_{\mathit{U}1}\right]$
Last 30 days	[115, 152]	[597, 1483]	[0.1923, 0.1024]
Last 365 days	[15679, 19150]	[37143, 33514]	[0.4221, 0.5713]

and

Table 2. US monthly records.

Period	Low Min, Low Max	High Min, High Max	${\mathit{F}}_{\mathit{N}1}\mathit{ϵ}\left[{\mathit{F}}_{\mathit{L}1},{\mathit{F}}_{\mathit{U}1}\right]$
Last 30 days	[0, 1]	[14, 25]	[0, 0.0384]
Last 365 days	[586, 985]	[2384, 2363]	[0.2457, 0.4166]

, respectively (see Aslam [50]). The weather data in intervals can be adequately analyzed using the proposed F-test as compared to the F-test under classical statistics. Table 1 and Table 2 indicate that the record-breaking data were recorded for the same time periods (daily records/monthly records); therefore, the statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ can be suitably applied when $\left(t_1=t_2\right)$. In Table 1 and Table 2, the values of $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ are also shown. The proposed test for the daily records (last 30 days) was implemented according to the following steps:

Step-1 

State $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ vs. $H_{1N}:{\mu }_{1N}\ne {\mu }_{2N}$.

Step-2 

Let $\alpha$= 5% and the corresponding critical value at $F_{\left[\left\{1196,230\right\},\left\{2968,304\right\}\right]}$ be 1 using the F-table from Kanji [1].

Step-3 

For the last 30 days, the values of $F_{N1}$ are calculated as $F_{N1}=\frac{\left[115,597\right]}{\left[152,1484\right]}; F_{N1}ϵ\left[0.1923,0.1024\right]$.

Step-4 

Accept $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ as $F_{N1}ϵ\left[F_{L1},F_{U1}\right]<1$.

The computed value of $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ falls in the acceptance region, which leads to acceptance of the null hypothesis with the conclusion that there is no significant difference between the two counts for less than 30 days. Similarly, the other results for US daily and monthly records can be interpreted.

4. Comparative Study

A statistical test based on neutrosophic statistics has an edge over the statistical test based on classical statistics. Therefore, in this section, a comparison of the proposed F-test with the existing F-test is presented. The proposed F-test consists of two parts known as the determinate and indeterminate parts, with the latter having the measure of indeterminacy. The proposed F-test is reduced to the existing F-test when record data are measured as exact values. The neutrosophic analysis for the last 30 days and the last 365 days is shown in

Table 3. Neutrosophic analysis of $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ two datasets.

Period	Neutrosophic Form	Measure of Indeterminacy
U.S Daily Records
Last 30 days	$F_{N1}=0.1923-0.1024I_{N130D}$	$I_{N130D}\left[0, 0.87\right]$
Last 365 days	$F_{N1}=0.4221+0.5713I_{N1365D}$	$I_{N13650D}\left[0, 0.2611\right]$
U.S Monthly Records
Last 30 days	$F_{N1}=0+\mathrm{0.0.084}{I}_{N130M}$	$I_{N130M}\left[0, 1\right]$
Last 365 days	$F_{N1}=0.2457+0.4166I_{N1365M}$	$I_{N1365M}\left[0, 0.4102\right]$

(see Aslam [50]).

For example, the neutrosophic form of the statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ for the records of the last 365 days (US daily record) is expressed as $F_{N1}=0.4221+0.5713I_{N1365D};I_{N1365D}\left[0, 0.2611\right]$. This neutrosophic form has two parts. The first part 0.4421 denotes the values of statistics under classical statistics. The second value $0.5713I_{N1365D}$ denotes the indeterminate part, and $I_{N1365D}$= 0.2611 is the measure of indeterminacy associated with the test. Note here that the proposed F-test is reduced to the F-test under classical statistics when $I_{N1365D}$= 0. From the analysis, it can be noted that the proposed F-test provides the values of the test statistic in the interval from 0.4221 to 0.5713 rather than the exact value. From this study, it is clear that the existing F-test gives the determinate value of the statistics while the proposed F-test for two counts from the Poisson distribution gives the values in the interval, in addition to information about the measure of indeterminacy.

Another advantage of the proposed test over the existing F-test is that, when $\alpha$= 5%, the probability of rejecting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ when it is true is 0.05, the probability of accepting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ is 0.95, and the measure of uncertainty/indeterminacy associated with the test is 0.2611. This is the case of paraconsistent probability where the total probability (0.05 + 0.95 + 0.2611 = 1.2611) lies in intervals from 1 and 3 (see Smarandache [37]). For the proposed test, the total probability associated with the test can be greater than 1; however, for the existing test, the total probability associated with the test is always equal to 1. Therefore, the proposed test is more informative than the existing F-test in terms of the probability associated with the test.

5. Power of the Test

In this section, we compare the performance of the proposed test with the existing F-test for counts in terms of the power of the test by following Aslam [51]. Suppose that $\alpha$ and $\beta$ are the type-I error (the chance of rejecting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ when it is true) and type-II error (the probability of accepting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ when it is false), respectively. Let $\left(1-\beta \right)$ be the power of the test. A simulation study was performed to calculate the power of the test for various values of $I_{F_{N1}}$. The values of the power of the test of the proposed F-test for counts when $\alpha$= 0.05 are shown in

Table 4. The values of the power of the test when $\alpha$ = 0.05.

${\mathit{I}}_{{\mathit{F}}_{\mathit{N}1}}$	$\left(1-\mathit{\beta }\right)$
[0, 0]	[0.90, 0.90]
[0, 0.2]	[0.90, 0.91]
[0, 0.4]	[0.90, 0.92]
[0, 0.6]	[0.90, 0.97]
[0, 0.8]	[0.90, 0.96]
[0, 1]	[0.90, 0.97]

. The values of the power of the test of the proposed F-test for counts when $\alpha$= 0.01 are shown in

Table 5. The values of power of the test when $\alpha$ = 0.01.

${\mathit{I}}_{{\mathit{F}}_{\mathit{N}1}}$	$\left(1-\mathit{\beta }\right)$
0	[0.86, 0.86]
0.2	[0.86, 0.89]
0.4	[0.86, 0.89]
0.6	[0.86, 0.91]
0.8	[0.86, 0.92]
1	[0.86, 0.94]

. The simulation process to calculate the power of the proposed test is explained as follows:

• Generate 100 values of counting data at various values of $I_{F_{N1}}$.
• Specify the values of $\alpha$= 0.01 and $\alpha$= 0.05 and select the corresponding table values using the F-table from Kanji [1].
• Compute the test statistic $F_{N1}ϵ\left[F_{L1},F_{U1}\right]$ and record the number of values accepting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ and not accepting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$.
• The power of the test $\left(1-\beta \right)$ can be computed from the ratio of values accepting $H_{0N}:{\mu }_{1N}={\mu }_{2N}$ to the total number of replications.

In Table 4 and Table 5 the first value when $I_{F_{N1}}=0$ presents the power of the test under classical statistics. From Table 4 and Table 5, it can be seen that, as the values of $I_{F_{N1}}$ increased, the values of the power of the test also increased. For example, from Table 4, it can be noted that the value of the power of the test was [0.90, 0.90] when $I_{F_{N1}}=0$ and the value of the power of the test was [0.90, 0.97] when$I_{F_{N1}}=0.6$. The curves of power of the proposed F-test for counts are shown in Figure 1 and Figure 2. Figure 1 and Figure 2 also show that as the values of $I_{F_{N1}}$ increased, the values of the power of the test also increased.

6. Concluding Remarks

The modification of the F-test for two counts of data came from the Poisson distribution presented in the paper. The mathematical proofs to the equations under neutrosophic statistics were added. The theoretical and application studies showed that the proposed F-test for two counts of data is a generalization of the existing F-test. From the analysis, it was noted that the proposed F-test performed better than the existing F-test in terms of information and probabilities associated with the test. The proposed F-test can be extended using a double sampling scheme for future research.