Among the fruit species of economic importance for Brazil, the most important are those belonging to the genus Spondias, represented by red mombin, otaheite apple, Brazil plum and yellow mombin, with relevance to mombin, due to their multiplicity of use and agroindustrial potential  . In Brazil, mombin plants are found mainly in the North and Northeast States, where their fruits receive different denominations, such as cajá, cajá true, cajá-mirim or taperebá. participation in the agribusiness of the Northeast region, mainly by commercialization for consumption as fresh fruit and pulp processing, which has great acceptance in the market for its exotic flavor, excellent quality and commercial value as raw material in the preparation of juices, popsicles, ice creams, nectars and jellies  .
Due to the nutritional value, associated to the economic and social importance of the mombin, several works have been carried out in the last years culminating with the selection of some clones, which were evaluated regarding the characteristics of the plants  and fruits   . However, despite the real and potential importance of this fruit, there is little technological information about it  .
In agricultural researches, the sizing of the sample required for the estimation of the mean of a given character is important when the population can not be measured, demands excessive time, financial and human resources  . The sample size is directly proportional to the variability of the data and the desired reliability in the estimation, being inversely proportional to the estimation error allowed a priori by the researcher    . The larger the sample size, the greater the precision of the experiment, with a reduction in the sample mean-variance, although the demand for resources is also high. On the other hand, small sample size may reduce experimental accuracy  .
In this way, the sample size for fruit characters has been determined for several crops such as melon  , orange fruit  , peach   , apple  , pineapple  , passion fruit  and papaya  .
However, studies that determine the sample size needed to evaluate physical-chemical characteristics of fruits of Spondias mombin L. were not found in the literature, despite the existence of papers that p ortray the quantification of their physical-chemical characteristics      .
Thus, the objective was to determine the sample size required to evaluate seven physical-chemical characters in mature fruits of Spondias mombin L.
2. Material and Methods
The fruits of Spondias mombin L. used in this work were harvested in an orchard located at Santa Angélica ranch (19.357˚ S latitude, 40.067˚ W longitude and 40 m altitude), in the municipality of Linhares, in the northern state of Espírito Santo, Brazil. Fruits of native plants were harvested in a shading area at maturation stage 4 (100% of the yellow bark) according to  , in the morning and transported to the laboratory for plant breeding.
The selection of 100 fruits was carried out after the elimination of those damaged or imperfect, which were evaluated in terms of characters: LL (longitudinal length, in mm); ED (equatorial diameter, in mm); FM (fruit mass, in g); PY (pulp yield, in%); TSS (total soluble solids, determined by bench refractometer and expressed in ˚Brix); TA (titrable acidity), determined by volumetric titration with indicator using NaOH 0.1 M and expressed as percentage of citric acid mass per volume of pulp in m/v, as described by the  ; and ratio (TSS/TA).
From the data of the characters evaluated in the 100 fruits of Spondias mombin L., the descriptive statistics (mean, minimum, maximum, standard deviation and coefficient of variation) were calculated. To verify the possibility of using the Student t distribution in the estimation of the optimal sample size, we tested the hypotheses regarding to asymmetry (Ho: asymmetry = 0, by the t test with p = 0.05), to Kurtosis (Ho: kurtosis = 3, by the t-test with p = 0.05) and adherence to the normal distribution by the Lilliefors test (p = 0.05)  .
The number of fruits to estimate the parameters of an infinite population to a desired level of accuracy was based on the half-range of the confidence interval for the mean, pelaEquação 1   :
On what: n is the sample size (number of fruits); is the critical value of the Student t distribution, whose area the right is equal to with degrees of freedom, and with 5% probability of error; s is the sample standard deviation; e is the relative error in the mean estimate (e = 0.01; 0.02; 0.04; 0.04; 0.05; 0.06; 0.07; 0.08; 0.09 and 0.10); m is the sample arithmetic mean.
To know the percentage error (e%) around the average, committed with the use of 100 fruits as simple, was used Equation (2)  :
Statistical analyses were performed using software R  .
3. Results and Discussion
Measures of central tendency, variability, asymmetry and kurtosis, and the Lilliefors test of the evaluation of the seven characters of mature fruits of Spondias mombin L. are presented in Table 1. All the characters presented a normal distribution of the sample data, although the fruit mass (FM) and the ratio showed significant asymmetry by the t test (p < 0.05) and the FM showed significant kurtosis by the t test (p < 0.05). Therefore, in relation to normality, it can be inferred that the data of the seven characters offer credibility to the study of sample size based on Student’s t distribution  .
The evaluated fruits presented the following means: 33.79 mm for longitudinal length (LL); 24.91 mm for equatorial diameter (ED); 13.35 g for fruit mass (FM); 61.67% pulp yield (PY); 1.70% titratable acidity (TA); 10.91 ˚Brix of total soluble solids (TSS); 6.56 ratio. These average values are close to those found for mature fruits of yellow mombin fruits by    , with the exception of the ratio values, which were slightly lower. This difference can be explained by the variability among the fruits harvested in this work since they were from seminiferous propagation plants, while the authors worked with clones selected for higher TSS values.
The magnitude of the coefficient of variation (CV%) ranged from 11.62% for yield (PY) to 32.02% for FM (Table 1). In the evaluation of pineapple fruit
Table 1. Mean, minimum, maximum, standard deviation (SD), coefficient of variation (CV%), asymmetry (AS), kurtosis + 3 (KT) and normality test results (Lilliefors, L) for eight characters measured in 100 mature fruits by Spondias mombin L.
(1)LL = longitudinal length, in mm, ED = equatorial diameter, in mm, FM = fruit mass, in g, PY = pulp yield, in %, TSS = total soluble solids, in ˚Brix, TA = titrable acidity, in % of citric acid and ratio = TSS/T). (2)Asymmetry differs from zero by the t-test, at a 5% probability level. ns = non-significant. (3)Kurtosis differs from three by t-test, at a 5% probability level. ns = non-significant. (4)S = Normal distribution by the Lilliefors test (P > 0.05).
characters,  also obtained higher CV for fruit mass. This suggests that, in order to obtain the mean estimate, with a certain precision, the sample size of PY and FM are the lowest and the highest, respectively, among the evaluated characters.
The sample size for the mean estimation, with an estimation error equal to 1% of the mean is 4028 fruits for the FM character (Table 2). High value for sample size has already been verified in fruit size sample studies for other crops, such as in peach, where 4213 fruits are required to estimate pulp firmness after refrigerated storage with 1% of mean estimation error  . These high sample sizes show that in practice, 1% sampling is infeasible because the estimated sample size may be larger than the number of observations available for evaluation.
In this study, with the use of 100 fruits, the greatest relative error was 6.35% for FM (Table 2). For this case, a good solution may be to admit a greater relative error as done by  in the character evaluation of yellow passion fruit. These authors assumed a 10% error. With this same percentage of error, 40 yellow mombin fruits are needed for determination of FM. With the same 40 fruits, the relative error will be less than 5% for LL, ED, PY and TSS. If the researcher assumes a 5% error, the following sample sizes (number of fruits) are required: LL (28); ED (29); FM (161); PY (21); TA (43); TSS (33); Ratio (68). The results were expected, because the sample size decreases when the error allowed around the mean increases (for example, from 5% to 10%). which is perceived for the FM character, where for a 10% error we have 40 fruits, whereas a 5% error is necessary 161 fruits.
The occurrence of variability of sample size for different characters measured in fruits has been reported for peach   , pineapple  , apple  and papaya  .
Table 2. Sample size (number of fruits) for the estimation of the average of eight characters of fruits of Spondias mombin L. harvested mature, for the relative errors of estimation equal to: 0.01; 0.02; 0.03; 0.04; 0.05; 0.06; 0.07; 0.08; 0.09; 0.10 of the mean estimate, and error of estimation as a percentage of the mean (e%) estimate, based on 100 evaluated fruits.
(1)LL = longitudinal length, in mm, ED = equatorial diameter, in mm, FM = fruit mass, in g, PY = pulp yield, in %, TSS = total soluble solids, in ºBrix, TA = titrable acidity, in % of citric acid and ratio = TSS/T).
Thus, when planning an experiment to be conducted with yellow mombin fruits, under conditions similar to those adopted in this study, in a completely randomized experimental design, for the estimation of the mean of each treatment with 10% accuracy, 40 fruits per treatment should be evaluated. If the experiment was planned with five replicates per treatment, eight fruits per replicate (40/5 = 8) would be sampled, that is, eight fruits per plot. Also, if the experiment were evaluated four treatments, the researcher would have to use 160 fruits to perform such experiment (4 × 40 = 160 fruits per treatment).
It should be noted that the sample size presented in this paper (Table 2) is for fruits harvested from seminiferous propagation plants and reflects a practice that is usual in extractive exploitation of mombin in Brazil, especially in the Northeast region  . The sample size required may be lower than those found here when the fruits come from clonal selections since Spondias mombin L. is a species with high genetic variability when propagated by seeds  .
For experimental evaluation of yellow mombin fruits considering an accuracy of 10% around the mean, 40 fruits per treatment should be evaluated, considering a completely randomized experiment.
The authors would like to thank CNPq and CAPES for their support in granting scholarships to authors of this publication.
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