Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 84 https://internationalpubls.com A Novel Method for the Fuzzy Survival and Fuzzy Hazard Models of the Corticosterone Effect Senthilkumar P1*, Shiny Epciya J1, Vijayabalan D1, Naresh kumar jothi2, Sukumaran D1 Balamurugan K3, Suresh G1 1Department of Mathematics, Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Avadi, Chennai- 600062. 2Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Avadi, Chennai-600062. 3Department of Mathematics, Dhanalaksmi Srinivasan University, Trichy-621112. Corresponding Author: psenthil9159115957@gmail.com Article History: Received: 23-07-2024 Revised: 10-09-2024 Accepted: 30-09-2024 Abstract To ascertain the significance of corticosterone release scores, theoretical research was conducted. We offer formulas for fuzzy Two-Parameter Weibull distributions, fuzzy Survival and Hazard functions, and related alpha-cut sets. We showed that, using fuzzy survival and hazard models based on two-parameter Weibull distributions, the smaller fuzzy survival model for the impact of corticosterone release increases and the higher fuzzy hazard model for the impact of corticosterone release decreases as the experiment termination alpha value increases. When the alpha value of the test's termination grows, the greater fuzzy survival scenario for the effect of corticosterone release decreases while the smaller fuzzy hazard models with the same effect increase. Keywords: Fuzzy Survival Model, Hypothesis testing, Log-Logistic distribution, Exponential distribution, Weibull distribution. 1. Introduction The use of the theory of fuzzy sets for fuzzy system analysis of survival has been the subject of numerous studies. The survival functions are the ones that are most frequently employed in lifetime data analysis. This function calculates the likelihood that a device will perform properly for a specific period of time. Numerous techniques and models used in traditional survival theory use the assumption that all lifespan density function parameters are accurate. However, in real life, randomness as well as fuzziness is combined over the system's lifetime. Zadeh [9] proposed fuzzy set theory in 1965. Later, fuzzy set theory and mathematics were developed and used in a variety of scientific domains. The authors of Chen, on the other C.H. and others [3], Chen, S.M. is [4], the authors of Cai et al. [6], as well as [7], which adjusted the system's assumptions and precisely characterized its success or failure based on the fuzzy indicate assumption, introduced and developed the concept of fuzzy Survival. The system may be in either the fuzzy success state possibility assumption or as the fuzzy failure level possibility assumption at any one time. The actions of a system can be fully described by possibility measurements. An introduction to systems failures and its use of fuzzy technique was provided by Cai [7]. In [6] and [7], a method for fuzzy system survival analysis using fuzzy numbers was presented. A technique of fuzzy system estimation of survival as well as alpha-cut operation on fuzzy numbers was presented by Chen S.M. [4]. Fuzzy survival function mathematical models that depend on the the Weibull distribution were described by Zdenek Karpisek as well as others [10]. For the investigation of fuzzy reliability in diverse systems, Utkin et al. [8] established a set of functional equations. The fuzzification value of 0.5 is the cross-over point. Any fuzzy value greater than 0.5, implies that the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 85 https://internationalpubls.com original phenomenon’s value may be a member of the set. It's probable that the original phenomenon's result is a member of the set when the fuzzification values decrease below 0.5. Potentially, the values are not a component of the set [1], [2]. The Section 2 of the text, which was set up as follows, is where the introductory data that has been employed on this article was presented; Log-Logistic distribution, Exponential distribution and Weibull distribution were used to introduce the various types of fuzzy mathematical models in Section 3. We were able to assess the impact of corticosterone on section 4 by employing the models previously addressed and calculating the rate of survival and the hazard values. In Section 5, using hypothesis testing, we investigate the chance of survival and hazard estimates of the various scenarios. Section 6 provides a succinct conclusion. Notation  - Scale parameter of LLD  - Shape parameter of LLD  - Scale parameter of WD  - Shape parameter of WD  - Scale parameter of ED  - Shape parameter of ED ][ H Sx - Alpha cut of Scale value in GGD ][ H Sx - Alpha cut of Shape value in GGD ][ H Sx - Alpha cut of Scale value in LLD ][ H Sx - Alpha cut of Shape value in LLD ][ H Sx - Alpha cut of Scale value in RD and GRD ][ H Sx - Alpha cut of Shape value in RD and GRD ][ H SxS - Survival Value of X ][ H SxH - Hazard Value of X ][ H SxS - Fuzzy Survival Value of X ][ H SxH - Fuzzy Hazard Value of X 2. Preliminaries 2.1 Definition: Assume that the lifetime H SX is a random variable that is continuous with the cumulative hazards function ( )H SF as well as  ),0 the interval hazard function ( )H Sf . Its survival mechanism is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 86 https://internationalpubls.com ( ) ( )   ( )   dfPFS H S H S H S H S   ==−=1 2.2 Properties: ➢ Every Survival function ( )H SS is monotonically decreasing, ( ) ( )   forallSS H S H S , ( )H SS ➢ Since CDF is a right-continuous function, the survival function ( ) ( ) H S H S FS −=1 is also right-continuous. ➢ The Expected Survival time ( ) ( )  dS H S  = 0 ➢ The density function of probability ( )H Sf as well as the hazard function ( )H SH can be connected to the survival function. ( ) ( )( )   H S H S S d d f −= ( ) ( )( )   H S H S S d d H log−= So that ( ) ( ) 1 0 1exp   dHS H S H S         −=  2.3 Definition: A hazard is a risky phenomena, substance, behavior, or circumstance. It might result in environmental harm, the loss of life, injuries, or other adverse health effects, as well as property damage, loss for livelihoods and services, social unrest, and economic upheaval. 2.4 Definition: Chronic stress, also referred to as ongoing, unresolved stress, is another name for survival mode. Every human has undergone some level of stress at some point, yet while in a state of survival, stress is so intense that it makes it difficult to unwind. The brain's fear-related regions are hyperactive. 2.5 Definition: The hazard function, abbreviated h, or hazard rate, is created by computing the failure rate in progressively smaller time intervals. As gets closer to zero, the following becomes the instant failure rate, or as we say instant hazard rate: ( ) ( ) ( ) ( )    H S H S H SH S R RR H * lim 0  +− = → 2.6 Definition: The very existence of the failure distribution, ( )H SF , and that is a cumulative distribution function that expresses the likelihood for failure (at least) up to and beyond period  , is a prerequisite for a continuous failure rate. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 87 https://internationalpubls.com   ( ) ( ) H S H S H S RFP −== 10,  - Stands for the failure rate. The failure density function's integral represents the failure distribution ( )H Sf ( ) ( )   dfF H S H S = 0 ( ) ( ) ( ) ( )    H S H S H S H SH S R f F f H = − = 1 2.7 Log -Logistic Distribution The parameter 0 is a scale parameter and is also Median of the distribution. The parameter 0 is a shape parameter. The distribution is unimodel when 1 and its dispersion decreases as  increases. The cumulative distribution function is ( )     −       + = 1 1 ,:F ( )              +       = 1 ,:F ( ) ( ) ( ) ( )     + =,:F Where 0,0,0   The probability density function of Log-Logistic Distribution is 1,0,0, 1 ),:( 2 1                +             = −           f The Survival function of Log -Logistic distribution is 1 1)( −               +=   H SXS )(),(,   The Hazard function of Log -Logistic distribution is               +             = − 1 )( 1 H SXH )(),(   Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 88 https://internationalpubls.com 2.8 Exponential Distribution A continuous random variable H SX with Exponential distribution ( ) , where, 0 is shape parameter and 0 is scale parameter has the probability density function is given by 0,0,0,),:( )(1 = −−−  ef H S The following gives the formula for the two-parameter exponential cumulative density function ( ) )(1),:(  −−−== eQF H S H S The Survival function of Log -Logistic distribution is ( ) 0,0, = − eXS H S H S The Hazard function of Log -Logistic distribution is ( ) 0,0, = H S H S XH 2.9 The Weibull distribution's probability density function A continuous random parameter H SX having a Weibull distribution with two parameters ( ) , has a probability density function, where 0 is the shape parameter and 0 is the scale parameter ( )         =       − −− 0,0,0,),:( 1      ef H S The following formulas provide the Weibull distribution's cumulative distribution function (CDF)           − −= eF H S 1),:( The Survival function of two parameter weibull distribution is .0,0,0,)( =       −     eS H S The Hazard function of two parameter weibull distribution is .0,0,0,)( =       −     eH H S 3. New finding 3.1 Fuzzy Expected Value and Fuzzy Variance Value of Fuzzy Log-Logistic Distribution Model A random variable H SX follows Fuzzy Log-Logistic Distribution is denoted by H SX ~ ),,( FLLD , where and  are fuzzy parameters The Survival value of H SX ~ ),,( FGRD is given by   === )(],)[(],)[(])[(|],)[()( )()( H SXS H SU H SL H SXS H S H S XXSXSXSXSXS H S H S Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 89 https://internationalpubls.com  )(),(|)(])[(  = H S H SL XSInfXS  )(),(|)(])[(  = H S H SU XSSupXS 1 1)( −               +=   H SXS , )(),(   The Hazard value for H SX ~ ),,( FLLD is given by   === )(],)[(],)[(])[(|],)[()( )()( H SXH H SU H SL H SXH H S H S XXHXHXHXHXH H S H S  )(),(|)(])[(  = H S H SL XHInfXH  )(),(|)(])[(  = H S H SU XHSupXH               +               = − 1 )( 1 H SXH , )(),(   3.2 Fuzzy Survival Value and Hazard Value of Fuzzy Exponential Distribution A continuous random variable H SX with Exponential distribution ( ) , where, 0 is shape parameter and 0 is scale parameter has the probability density function is given by 0,0,0,),:( )(1 = −−−  ef H S . The following gives the formula for the two-parameter exponential cumulative density function ( ) )(1)(   −−−== tetQtF A random variable H SX as follows Fuzzy Exponential distribution (FED) with the fuzzy numbers  , is indicated by H SX ~ ).,,( FED The Survival value for H SX ~ ),,( FED is    === )(],)[(],)[(])[(|],)[()( )()( H SXS H SU H SL H SXS H S H S XXSXSXSXSXS UH S  )(),(|)(])[(  = H S H SL XSInfXS  )(),(|)(])[(  = H S H SU XSSupXS ( ) 0,0, = − eXS H S H S The Hazard value for H SX ~ ),,( FED is    === )(],)[(],)[(])[(|],)[()( )()( H SXH H SU H SL H SXH H S H S XXHXHXHXHXH UH S Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 90 https://internationalpubls.com  )(),(|)(])[(  = H S H SL XHInfXH  )(),(|)(])[(  = H S H SU XHSupXH ( ) 0,0, = H S H S XH 3.3 Fuzzy Survival Value and Hazard Value of Fuzzy Weibull Distribution We consider the Weibull distribution with fuzzy parameters by replacing the scale parameter  into the fuzzy number  and shape parameter  into  The Fuzzy Probability density function of Weibull Distribution is ( )           =         − −       ef 1)(,: A random variable H SX follows Fuzzy Weibull Distribution is denoted by H SX ~ ),,( FWD , where and  are fuzzy parameters The Survival value of H SX ~ ),,( FWD is given by   === )(],)[(],)[(])[(|],)[()( )()( H SXS H SU H SL H SXS H S H S XXSXSXSXSXS H S H S  )(),(|)(])[(  = H S H SL XSInfXS  )(),(|)(])[(  = H S H SU XSSupXS 1 )( −         − =    eXS H S , )(),(   The Hazard value for H SX ~ ),,( FWD is given by   === )(],)[(],)[(])[(|],)[()( )()( H SXH H SU H SL H SXH H S H S XXHXHXHXHXH H S H S  )(),(|)(])[(  = H S H SL XHInfXH  )(),(|)(])[(  = H S H SU XHSupXH 1 )( −         − =    eXH H S , )(),(   4. APPLICATION Let us consider an example for concentration of Corticosterone were determined in blood samples of rat, with free access to food and water under the condition of constant temperature and fixed 12- hours light/12-hours dark photoperiod (light on from 07.30 am to 19.30 hours) for at least two weeks prior to surgery. During this time, the rats were accustomed to the presence of the experimenter by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 91 https://internationalpubls.com daily handling. The experiments were carried out in early spring. The effects of Corticosterone release in rats were measured [5]. Fig. 4.1 Corticosterone releases of rats over a 24- hour light/dark period. In some situations the value of the scale and shape parameters of the two parameter Weibull distribution are not known precisely. Therefore we consider triangular numbers for the scale and shape parameter. The triangular fuzzy number of the scale and the shape parameters respectively are ]5.2,106.2,2[= and ]4,519.3,5.3[= . The alpha cut of scale, shape and location parameters respectively are ]394.05.2,106.02[][  −+= ]481.04,519.05.3[][  −+= Table 4.1: α-cut of Fuzzy Survival Model α FLLD FED FWD FLLD FED FWD Lower α-cut Upper α-cut 0 0.99981986 1 1 0.99999999 1 1 0.1 0.99981964 0.995325 0.999997378 0.99999998 0.987465 1 0.2 0.99982008 0.989805 0.999987331 0.99999997 0.975936 1 0.3 0.99982016 0.983449 0.99996728 0.99999986 0.965383 1 0.4 0.99982024 0.976267 0.999925906 0.99999975 0.955776 0.99999997 0.5 0.99927248 0.968272 0.999833277 0.99999777 0.947089 0.999999238 0.6 0.99789587 0.959479 0.99961079 0.99998588 0.939298 0.999989156 0.7 0.99513548 0.949907 0.999041833 0.99993168 0.932382 0.999896236 0.8 0.99040435 0.939574 0.997498483 0.99972833 0.926323 0.999257671 0.9 0.98317475 0.928502 0.993071629 0.99906998 0.921105 0.99575206 1 0.9730519 0.916714 0.979721037 0.99717291 0.916714 0.979721037 Table 4.2: Lower α-cut of Fuzzy Hazard Model α FLLD FED FWD FLLD FED FWD Lower α-cut Upper α-cut 0 0 0 4 0 0 14.1 0.1 0.00000002 0.004686 5.007686868 0 0.012614 14.0977 0.2 0.00000325 0.010247 6.015323792 0.00000002 0.024358 14.0954 0.3 0.00005324 0.01669 7.022870204 0.00000068 0.03523 14.09309999 0.4 0.00033664 0.02402 8.030204969 0.00000884 0.045232 14.09079958 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 92 https://internationalpubls.com 0.5 0.00127952 0.032243 9.036993075 0.0000662 0.054363 14.08848927 0.6 0.00355434 0.041364 10.04228992 0.00035026 0.062623 14.08604725 0.7 0.00799979 0.051391 11.04330851 0.00145854 0.070012 14.0824386 0.8 0.01550086 0.062329 12.03142771 0.00509729 0.076532 14.07114682 0.9 0.0268727 0.074183 12.97875104 0.01557722 0.082181 14.01949198 1 0.04277145 0.08696 13.79153303 0.04277145 0.08696 13.79153303 Fig: 4.2 Lower α-cut of Survival Model Fig: 4. 3 Upper α-cut of Fuzzy Survival Model Fig: 4.4 Lower α-cut of Fuzzy Hazard Model Fig: 4. 5 Upper α-cut of Fuzzy Hazard Model 5. Testing of Hypothesis: Testing of hypotheses is a procedure used to determine the degree of trial validity and provides a strategy for population-related decision-making, i.e., it conveys a method for acknowledging the consistency with which one can extrapolate experimental results from the sample under examine to the larger population that from which the population being studied was drawn. We start by defining a hypothesis, which is a specific statement of the population's parameters. An example of such a hypothesis is 0H . Here, we define 0H in the following manner: 0: 210 − InfInfH  There is significant difference in 1Inf than 2Inf 0: 211 − InfInfH  Test statistics for Lower alpha values is defined by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 93 https://internationalpubls.com               − + − − = 11 2 2 2 1 2 1 21 Inf Inf Inf Inf InfInf Inf nn    ( )         − − =  1 12 1 Inf InfInf Inf n   and ( )         − − =  12 212 2 Inf InfInf Inf n   Test statistics for Upper alpha values is defined by               − + − − = 11 2 2 2 1 2 1 21 Sup Sup Sup Sup SupSup Sup nn    ( )         − − =  11 12 1 Sup SupSup Sup n   and ( )         − − =  12 22 2 Sup SupSup Sup n   5.1 Lower Fuzzy Survival Null hypothesis :0LLELH The LFS in FLLD and FED do not differ much from one another. Alternative hypothesis 211 : LLELH Null hypothesis :0EWLH The LFS among FED and FWD does not significantly differ. Alternative hypothesis 311 : EWLH Null hypothesis :0WLLLH The LFS from FWD and FLLD is not significantly different from each other. Alternative hypothesis 321 : WLLLH Table 5.1 Calculation of Sample Means and Standard Deviations of Lower Fuzzy Survival α X1 X2 X3 S1*S1 S2*S2 S3*S3 0 0.99981986 1 1 0.0419512 0.01565001 0.0001664 0.1 0.99981964 0.995325 0.999997378 0.0419511 0.014502181 0.0001663 0.2 0.99982008 0.989805 0.999987331 0.0419513 0.013203159 0.0001661 0.3 0.99982016 0.983449 0.99996728 0.0419513 0.011782885 0.0001656 0.4 0.99982024 0.976267 0.999925906 0.0419513 0.010275269 0.0001645 0.5 0.99927248 0.968272 0.999833277 0.0417272 0.00871833 0.0001621 0.6 0.99789587 0.959479 0.99961079 0.0411667 0.007153607 0.0001565 0.7 0.99513548 0.949907 0.999041833 0.0400542 0.00562605 0.0001426 0.8 0.99040435 0.939574 0.997498483 0.0381829 0.004182726 0.0001081 0.9 0.98317475 0.928502 0.993071629 0.0354097 0.002873174 0.0000357 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 94 https://internationalpubls.com 1 0.9730519 0.916714 0.979721037 0.0317025 0.001748411 0.0000544 Calculated value of 124980659.3,538719181.0,099255.3 === WLLEWLLE ttt At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Calculated LLEt bigger than Tabulated LLEt The null hypothesis 0LLEH is rejected. 11+11-2=20 d. f. has a tabulated value of 2.080 at the 5% level of significance. Calculated value of EWt is Less than the value of EWt in the table. The null hypothesis 0EWH is accepted. At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Calculated value of LLWt is higher than the value LLWt of in the table. We do accept the null theory 0LLWH . 5.2 Upper Fuzzy Survival Null hypothesis :0LLEUH The UFS in FLLD and FED are not significantly different from one another. Alternative hypothesis 211 : LLEUH Null hypothesis :0EWUH This UFS for FED and FWD are identical, and this is a significant distinction. Alternative hypothesis 311 : EWUH Null hypothesis :0WLLUH Its UFS in the FLLD and FWD are identical, and this is a significant distinction. Alternative hypothesis 321 : WLLUH Table 5.2 Calculation of Sample Means and Standard Deviations of Upper Fuzzy Survival α Y1 Y2 Y3 S1*S1 S2*S2 S3*S3 0 0.99999999 1 1 0.0091585 0.00948676 0.45306361 0.1 0.99999998 0.987465 1 0.0091585 0.012085704 0.45306361 0.2 0.99999997 0.975936 1 0.0091585 0.014753503 0.45306361 0.3 0.99999986 0.965383 1 0.0091585 0.017428488 0.453063611 0.4 0.99999975 0.955776 0.99999997 0.0091585 0.020057357 0.453063651 0.5 0.99999777 0.947089 0.999999238 0.0091589 0.022593397 0.453064636 0.6 0.99998588 0.939298 0.999989156 0.0091612 0.024996242 0.453078208 0.7 0.99993168 0.932382 0.999896236 0.0091716 0.02723094 0.453203308 0.8 0.99972833 0.926323 0.999257671 0.0092106 0.02926734 0.454063484 0.9 0.99906998 0.921105 0.99575206 0.0093374 0.031079927 0.458800231 1 0.99717291 0.916714 0.979721037 0.0097076 0.032647431 0.480774387 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 95 https://internationalpubls.com Calculated value of 501450021.3,778690192.0,442251.3 === WLLUEWULLEU ttt For the 11+11-2=20 d.f., the tabulated value of LLEUt is 2.080 at the 5% level of significance. Value of LLEUt higher than calculated value of LLEUt in the table Rejected is the null hypothesis 0LLERUH . At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Calculated value of EWUt is bigger than the tabulated value of EWUt We reject the null hypothesis 0EWUH . 11+11-2=20 d.f. has a tabulated value of 2.080 at the 5% level of significance. Calculated value of WLLUt > Tabulated value of WLLUt We do not accept the null hypothesis 0WLLUH . 5.3 Lower Fuzzy Hazard Null hypothesis :0LLELVH The LFH in FLLD and RED do not differ much from one another. Alternative hypothesis 211 : LLELVH Null hypothesis :0EWLVH The LFH among FED and FWD does not significantly differ. Alternative hypothesis 311 : EWLVH Null hypothesis :0WLLLVH The LFH from FWD and FLLD is not significantly different from each other. Alternative hypothesis 321 : WLLLVH Table 5.3 Calculation of Sample Means and Standard Deviations of Lower Fuzzy Hazard α X1 X2 X3 S1*S1 S2*S2 S3*S3 0 0 0 4 0.632025 0.76545001 9.0775664 0.1 0.00000002 0.004686 5.007686868 0.632025 0.757272406 16.1651188 0.2 0.00000325 0.010247 6.015323792 0.6320198 0.74762481 25.2830345 0.3 0.00005324 0.01669 7.022870204 0.6319404 0.736524404 36.4305220 0.4 0.00033664 0.02402 8.030204969 0.6314899 0.723996774 49.6053276 0.5 0.00127952 0.032243 9.036993075 0.6299922 0.71007082 64.8007785 0.6 0.00355434 0.041364 10.04228992 0.6263862 0.694782263 81.9964644 0.7 0.00799979 0.051391 11.04330851 0.6193693 0.678167073 101.1273297 0.8 0.01550086 0.062329 12.03142771 0.6076189 0.66027163 121.9771745 0.9 0.0268727 0.074183 12.97875104 0.5900195 0.641147714 143.7996946 1 0.04277145 0.08696 13.79153303 0.5658478 0.620849444 163.9535053 Calculated value of 723051839.2,081386209.2,57484.1 === WLLVEWVLLDV ttt At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 96 https://internationalpubls.com Calculated LLELVt less than value LLELVt 's tabulated The null hypothesis 0LLEVH is acceptable. 11+11-2=20 d.f. has a tabulated value of 2.080 at the 5% level of significance. Calculated value of EWVt is bigger than the value of EWVt in the table. The null hypothesis 0EWVH is rejected. At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Calculated value of WLLLVt is higher than the value of WLLLVt in the table. We do not accept the null theory 0WLLVH . 5.4 Upper Fuzzy Hazard Null hypothesis :0LLEVLH The UFH in FLLD and FED are not significantly different from one another. Alternative hypothesis 211 : LLEVVLH Null hypothesis :0EWVLH This UFH for FED and FWD are identical, and this is a significant distinction. Alternative hypothesis 311 : EWVLH Null hypothesis :0WLLVLH Its UFH in the FWD and FLLD are identical, and this is a significant distinction. Alternative hypothesis 321 : WLLVLH Table 5.4 Calculation of Sample Means and Standard Deviations of Upper Fuzzy Hazard α Y1 Y2 Y3 S1*S1 S2*S2 S3*S3 0 0 0 14.1 1.2005585 1.20428676 154.4278436 0.1 0 0.012614 14.0977 1.2005585 1.176760666 154.3706852 0.2 0.00000002 0.024358 14.0954 1.2005584 1.151419134 154.3135373 0.3 0.00000068 0.03523 14.09309999 1.200557 1.128205109 154.2563998 0.4 0.00000884 0.045232 14.09079958 1.2005391 1.1070575 154.1992628 0.5 0.0000662 0.054363 14.08848927 1.2004134 1.087926183 154.1418907 0.6 0.00035026 0.062623 14.08604725 1.1997911 1.07076344 154.0812594 0.7 0.00145854 0.070012 14.0824386 1.1973644 1.055526103 153.9916844 0.8 0.00509729 0.076532 14.07114682 1.1894143 1.042171473 153.711565 0.9 0.01557722 0.082181 14.01949198 1.1666652 1.030669618 152.433395 1 0.04277145 0.08696 13.79153303 1.1086585 1.020988994 146.8564192 Calculated value of 158069152.3,82.41717709,551309.1 === WLLVLEWVLLLVL ttt For the 11+11-2=20 d.f., the tabulated value of LLEUt is 2.080 at the 5% level of significance. Value of LLEUt less than calculated value of LLEUt in the table Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 97 https://internationalpubls.com Accepted is the null hypothesis 0LLEVLH . At a 5% level of significance, the tabulated value of 11+11-2=20 d.f. is 2.080. Calculated value of EWVLt is bigger than the tabulated value of EWVLt We reject the null hypothesis 0EWVLH . 11+11-2=20 d.f. has a tabulated value of 2.080 at the 5% level of significance. Calculated value of WLLVLt > Tabulated value of WLLVLt We do not accept the null hypothesis 0WLLVLH . Table 5.5: Paired sample t-test for fuzzy Survival Model for the effect of Corticosterone Genera lized Raylei gh Test Calculated value Table Value Hypothesis d. f Result Lower Fuzzy Hazard Upper Fuzzy Hazard Lower Fuzzy Hazard Upper Fuzzy Hazard LLEVLt 0.4316 55 0.946061 2.086 Accepted Accepted 5% The Fuzzy Hazard in the Log- Logistic distribution and the Exponential distribution do not differ significantly from one another. EWVLt 1.1051 02 0.231919 2.086 Accepted Accepted The Fuzzy Hazard in the Exponential distribution and the Weibull distribution differ significantly. WLLVLt 0.0440 39 0.009859 2.086 Accepted Accepted The Weibull distribution's fuzzy Hazard and the Log-Logistic distribution differ significantly. LLEVLt 0.4316 55 0.946061 2.845 Accepted Accepted 1% The Fuzzy Hazard in the Log- Logistic distribution and the Exponential distribution do not significantly differ from one another. EWVLt 1.1051 02 0.231919 2.845 Accepted Accepted The Fuzzy Hazard in the Exponential distribution and the Weibull distribution do not differ significantly from one another. WLLVLt 0.0440 39 0.009859 2.845 Accepted Accepted The Fuzzy Hazard in the Weibull distribution and the Log-Logistic distribution are very different from one another. Table 5.6: Paired sample t-test for fuzzy Hazard Model for the effect of Corticosterone Test Calculated value Table Value Hypothesis d. f Result Lower Fuzzy Survival Upper Fuzzy Survival Lower Fuzzy Survival Upper Fuzzy Survival Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol X No. Y (20--) 98 https://internationalpubls.com LLEt 0.076463 0.101402 2.086 Accepted Accepted 5% The Fuzzy Survival in the Log- Logistic distribution and the Exponential distribution differ significantly from one another. EWt 3.278995 3.920347 2.086 Rejected Rejected The Fuzzy Survival in the Exponential distribution and the Weibull distribution do not differ significantly. WLLt 3.291040 3.931538 2.086 Rejected Rejected The Weibull distribution distribution's Fuzzy Survival and the Log-Logistic distribution differ significantly. LLEt 0.076463 0.101402 2.845 Accepted Accepted 1% The Fuzzy Survival in the Log- Logistic distribution and the Exponential distribution significantly differ from one another. EWt 3.278995 3.920347 2.845 Rejected Rejected The Fuzzy Survival in the Exponential distribution and the Weibull distribution do not differ significantly from one another. WLLt 3.291040 3.931538 2.845 Rejected Rejected The Fuzzy Survival in the Weibull distribution and the Log-Logistic distribution are very different from one another. 5. Conclusion: In this paper, Fuzzy Survival and Hazard model with two parameter weibull distribution for the effect of release of Corticosterone with different alpha values were discussed. Using two parameter weibull distributions, it is clear that the α-cut for the Lower fuzzy Survival and Upper fuzzy Hazard values increases for alpha value increases. Similarly the α-cut for the Upper fuzzy Survival and Lower fuzzy Hazard values decreases when alpha value increases. This shows that if the test termination alpha value increases, the Lower fuzzy Survival Model for the effect of release of Corticosterone increases and Upper fuzzy Hazard model for the effect of release of Corticosterone decreases and if the test termination alpha value increases, the upper fuzzy Survival Model for the effect of release of Corticosterone decreases and lower fuzzy Hazard model for the effect of release of Corticosterone increases.Also by estimating the Fuzzy Survival and the Fuzzy Hazard of FLLD, FED and FWD, we were able to successfully create the fuzzy model to calculate the effect of Corticosterone. Lower alpha cuts result in higher mean values, and for upper alpha cuts, lowered. The results of the testing of the hypothesis reveal a substantial difference between FLLD and FED, FED and FWD, FWD and FLLD. For assessing the impact of Corticosterone, FWD and FLLD work effectively. References [1] Beuving G Vooder gma. 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