Testing average wind speed using sampling plan for Weibull distribution under indeterminacy

The time truncated plan for the Weibull distribution under the indeterminacy is presented. The plan parameters of the proposed plan are determined by fixing the indeterminacy parameter. The plan parameters are given for various values of indeterminacy parameters. From the results, it can be concluded that the values of sample size reduce as indeterminacy values increase. The application of the proposed plan is given using wind speed data. From the wind speed example, it is concluded that the proposed plan is helpful to test the average wind speed at smaller values of sample size as compared to existing sampling plan.

Preliminaries. Suppose  where µ is true average wind speed and µ 0 is the specified average wind speed. Based on this information, the proposed sampling plan is stated as follows Step 1 Select a random sample of n number of days and record the daily average speed for these selected days. Specify the number of days, say c , average wind speed µ 0 and indeterminacy parameter I N ǫ[I L , I U ].
Step 2 Accept H 0 : µ = µ 0 if daily average wind speed in c days is more than or equal to µ 0 , otherwise, reject The proposed sampling scheme is characterized by the three parameters n , c and I N , where I N ǫ[I L , I U ] is considered as the specified parameter and set according to the uncertainty level. Suppose that t 0 = aµ 0 be the time in days, where a is the termination ratio. The probability of accepting H 0 : µ = µ 0 is given by where p is the probability of rejecting H 0 : µ = µ 0 and obtained using Eqs. (2) and (3) and defined by where µ/µ 0 is the ratio of true average daily wind speed to specified average daily wind speed. Suppose that ∼ α and ∼ β be type-I and type-II errors. The meteorologists are interested to apply the proposed plan for testing H 0 : µ = µ 0 such that the probability of accepting H 0 : µ = µ 0 when it is true should be larger than 1− ∼ α at µ/µ 0 and the probability of accepting H 0 : µ = µ 0 when it is wrong should be smaller than ∼ β at µ/µ 0 = 1 . The plan parameters for testing H 0 : µ = µ 0 will be obtained such that the following two inequalities are satisfied.  Tables 1, 2, 3, 4, 5 and 6. Tables 1 and 2 are shown for the exponential distribution case. For exponential distribution, it can be seen that the values of n decreases as the values of a increases from 0.5 to 1.0. On the other hand for other the same parameters, the values of n decreases as the values of β increases. Note here that the indeterminacy parameter I N also plays a significant role in minimizing the sample size.

Comparative study
In this section, the efficiency of the proposed plan is discussed in terms of sample size. The smaller the sample size means that less cost is needed for testing the hypothesis about the daily average wind speed. Note here that the proposed sampling plan is the generalization of the plan under classical statistics when no uncertainty/ indeterminacy is found in recording the daily average wind speed. The proposed sampling plan reduces to the existing sampling plan when I N =0. The first column in Tables 1, 2, 3, 4, 5 and 6 presents the plan parameters  under the classical statistics. From Tables 1, 2, 3, 4, 5 and 6, it can be noted that the values of the sample size required for testing H 0 : µ = µ 0 decreasing as the indeterminacy parameter I N increases. For example, when µ/µ 0 = 1.1 and a = 0.5 from Table 1, it can be seen that n = 1143 from the plan under classical statistics and n = 1010 for the proposed sampling plan when I N = 0.05 . From the study, it is concluded that the proposed plan under indeterminacy is efficient in sample size as compared to the existing sampling plan under classical statistics. Therefore, the application of the proposed plan for testing the null hypothesis H 0 : µ = µ 0 requires a smaller sample as compared to the existing plan. The meteorologist can apply the proposed plan under uncertainty with fewer effort and time.

Application for wind speed data
The application of the proposed sampling plan will be discussed using wind speed data. The wind speed is a big and important source of energy. Due to the randomness and uncertainty, the wind speed data follows the statistical distribution under neutrosophic statistics. The meteorologists are interested to see the daily average wind speed under indeterminacy. The average wind speed data of Cairo city is taken from 40 and shown in Table 7. It is found that the wind speed data follows the Weibull distribution with shape parameter β = 2.7857 and scale parameter α = 8.05 . The plan parameters for this shape parameter are shown in Tables 8 and 9. For the proposed plan, the shape parameter is β N = (1 + 0.02) × 2.7857 ≈ 3 when I U = 0.02 . Suppose that meteorologists are interested to test H 0 : µ = 7.15 with the aid of the proposed sampling plan when I U = 0.02 , α = 0.10 , µ/µ 0 = 1.3 , a = 0.5 and β = 0.25 . From Table 5, it can be noted that n = 100 and c = 7 . The proposed sampling plan will be implemented as: accept the null hypothesis H 0 : µ = 7.15 if average daily speed in 7 days is more than equal to 7.15mph. From the data, it can be noted average daily wind speed is more than equal to 7.15mph in more than 7 days, therefore, the claim about the average daily wind speed H 0 : µ = 7.15 will be accepted. From the real example, it is concluded that the proposed sampling will be helpful to check the daily average wind speed.    www.nature.com/scientificreports/

Concluding remarks
The time truncated plan for the Weibull distribution under the indeterminacy was presented. The plan parameters of the proposed plan were determined by fixing the indeterminacy parameter. The plan parameters were given for various values of indeterminacy parameters, shape parameter, and scale parameter. Several tables for the application of the proposed plan are given. The application of the proposed plan was given with the help of daily average wind speed. The testing of the hypothesis was done to test the average daily wind speed. From the study, it is concluded that the indeterminacy parameter plays a significant role in fixing the plan parameters. The less sample size is needed as the indeterminacy parameter increased. In addition, it is found that the proposed plan is efficient than the existing sampling plan in terms of sample size. To save time, efforts, and energy, it is recommended to apply the proposed plan for testing the average wind speed. The proposed plan can be applied in metrology, oceanography, and thermodynamics. The proposed plan can be applied for testing big data from oceanography as future research. By following 41,42 , the software for goodness of fit tests using the npdf in Eq. (1) can be developed as future research.