American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Sampling Distribution: Impact of the Population Reliability on the Sample Size Determination Dr. Banzi Wellars* University of Rwanda. College of Science and Technology. Department of Mathematics. Kigali city. PO BOX 3900. Rwanda Email: banziwe10@yahoo.fr Abstract One of the spiniest problems in the elementary theory of the sampling is the determination of the size of the sample. Several authors tried to answer this question. In this paper, we present the methods of determination of the sample size in order to estimate to a given precision the population mean µ , the variance of the population being known or unknown. In the same way, we present the methods of determination of the size of the sample in order to estimate to a given precision the frequency f of a character in the population at different level of its reliability denoted here by p .the method uses the sampling error of the distribution of the specified statistic. Notice that the bigger the population reliability ( 1≅ or 100%), the smaller is the necessary sample size to get the best estimation of the parameter. The representativeness of the population doesn't essentially depend on the size of the sample; the size of the sample influences the precision of the measures; to achieve a highest accuracy, it is necessary to increase the sample size. The results of this research will help in determining the sample size to be drawn from different populations (finite or finite). AMS (MSC 2000) Mathematics Subject Classification: 62D05, 62E99 Keywords: Sampling, Sample size, Estimation, Error 1. Introduction In sampling analysis the most ticklish question is: what should be the size of the sample or how large or small should be the sample size “ n ”? If the sample size “ n ”is too small, it may not serve to achieve the objectives and if it is too large, we may incur huge cost and waste resources. ------------------------------------------------------------------------ * Corresponding author. E-mail address: banziwe10@yahoo.fr. 143 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 As a general rule, one can say that the sample must be of an optimum i.e., it should neither be excessively large nor too small. Technically, the sample size should be large enough to give a confidence interval of desired width and as such the size of the sample must be chosen by some logical process before sample is taken from the universe. In determining the sample size in a statistical experiment, we must know two things: How close we wish our estimate to be to the true value of the population parameter, and how certain we wish to be that our estimate will be within the selected number of units of the value of the parameter [6]. Furthermore, size of the sample should be determined by a researcher keeping in view the following points [4]: Nature of universe: universe may be either homogeneous or heterogeneous in nature. If the items of the universe are homogeneous, a small sample can serve the purpose. But if the items are heterogeneous, a large sample would be required. Technically, this can be termed as the dispersion factor. Number of classes proposed: If many class-groups are to be formed, a large sample would be required because a small sample might not be able to give a reasonable number of items in each class-group. Nature of study: If items are to be intensively and continuously studied, the sample should be small. For a general survey the size of the sample should be large, but a small sample is considered appropriate in technical surveys. Type of sampling: Sampling technique plays an important part in determining the size of the sample. A small random sample is apt to be much superior to a large bad badly selected. Standard of accuracy and acceptable confidence level: If the standard of accuracy or the level of precision is to be kept high, we shall require relatively larger sample. For doubling the accuracy for fixed significance, the sample size has to be increased fourfold. Availability of finance: In practice, size of the sample depends upon the amount of money available for the study purposes. This factor should be kept in view while determining the size of sample for large samples result in increasing the cost of sampling estimates. Other considerations: nature of units, size of the population, size of questionnaire, availability of trained investigators or enumerators in case of structured interview, the conditions under which the sample is being conducted, the time available for completion of the study are few other considerations to which a researcher must pay attention while selecting the size of the sample [2,5]. There are two alternative approaches for determining the size of the sample. The first approach is “to specify the precision of estimation desired and then to determine the sample size necessary to insure it” and the second approach “uses Bayesian statistics to weight the cost of additional information against the expected value of the additional information”. The first approach is capable of giving a mathematical solution, and as such is frequently used technique of determining “ n ”. The limitation of this technique is that it does not analyze the cost of gathering information vis-a-vis the expected value of information. The second approach is theoretically optimal, but it is seldom used because of the difficult involved in measuring the value of information. Hence, we 144 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 shall mainly concentrate on the first approach [4]. 2. Determination of Sample size trough the approach based on precision rate and confidence level Researcher will have to specify the precision that he wants in respect of his estimates concerning the population parameters. For instance, a researcher may like to estimate the mean of the universe within “ 2± ” of the true mean with 95 per cent confidence. In this case we will say that the desired precision is 2± , i.e., if the sample mean is $100, the true value of the mean will be not less $98 and no more than $102. In other words, all this means that the acceptable error denoted here by d , is equal to 2. 2.1. Sampling Distribution of Mean 2.1.1. The variance of the population known Sample size when estimating a mean: the confidence interval for the universe mean µ is given by cX z n σ ± (2.1) Where X is the sample mean and cz the critical value of the standard normal variable corresponding the given confidence level. Suppose we have 4.8σ = for our purpose. If the difference between µ and X or the acceptable error is to be kept within 4± of the sample mean with 95% confidence, then we can express the acceptable error d as equal to P cd z n σ = × or 4.84 1.96 n = × hence ( ) ( ) ( ) 2 2 2 1.96 4.8 5.531904 6 4 n × = =  In a general way, if we want to estimate the mean µ in a population with standard deviation σ with an error no greater than " "d by calculating a confidence interval with confidence corresponding to z , the necessary sample size " "n is obtained by [2]: 2 2 2 zn d σ = (2.2) All this is applicable when the population happens to be infinite. But in case of finite population of size N , the above stated formula for determining sample size will become [4] 2 2 2 2 2( 1) z Nn N d z σ σ × × = − + (2.3) 145 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 2.1.2. The population variance unknown We considered the question of the determination of the number of measures previously while supposing known the standard deviation of the population. Actually this parameter is generally unknown or known pain, but often, the coefficient of variation can be considered fortunately like known or practically known. It is thus frequently for example in the biologic domain, because the errors of numerous variables are roughly proportional to the averages, so that the coefficients of variation are practically constant in conditions data. While always supposing true the conditions of the previous section, the gotten previously relation becomes Either  22 1 2 2 z n d α σ − = (2.4) or 2 2 1 2 2 z V n d α − = (2.5) In this last relation, V designates the coefficient of variation, expressed in percent of the average, and, d, the maximum relative mistake or the relative mistake margin, i.e., the mistake maximum corresponding rd to one degree of confidence level 1 α− and expressed also in percent of the average. For small values of n, the variable z doesn't follow the normal law anymore, but rather the law established by Student, the pseudonym of the English statistician W. G. Gosset that published in 1908, under this name, a survey carrying on this variable [1]. Then, the variable t following by definition a law of student with ν degrees of freedom is defined by the following probability density function 1 2 2 1 2( ) 1 2 tf t νν ν ννπ + − + Γ    = +    Γ    for t−∞ < < ∞ (2.6) Where Γ denotes gamma function and ν the number of degrees of freedom. There exist as many probability curves densities as of valuesν . Thus, for small values of n, the relation (2.5) is been able to convenient, because it must be used by successive approximation and, the formulas (2.5) become then, 2 2 1 2 2 t V n d α − = , where 1 2 t α − dependent approximation of n, through the intermediary of number of degrees of freedom. However one can give a graphic representation of it as abacus. Inversely this abacus can also serve to determine graphically, in relative value, the half-length of one interval of confidence, knowing the coefficient of 146 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 variation V and the efficient n [1, 5] Figure1: Relation between the number of n measures, the coefficient of V variation, and the d margin mistake relative rd , for a confidence level 95%, When n is sufficiently big, the quantity 1 2 t α − can be replaced by the corresponding value of the standard normal distribution. In these conditions, one has respectively for 0.05α = and 0.01α = : 2 2 4 r Vn d  and 2 2 7 r Vn d  (2.7) Because and 0.975 1.96 2z =  and 0.995 2.575 7z =  Example: The following table gives the weights of three shares of chickens, measures to three different dates: Averages, Standard deviations and Coefficient of variation estimated. 147 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 2.1: Averages, standard deviation and coefficient of variation of three shares of chicken Average Standard deviation Coefficient of variation (gr) (gr) V (%) 1,067 1,308 1,518 134 166 164 12.5 12.7 10.8 1,022 1,242 1,421 138 162 169 13.5 13.0 11.9 876 1,094 1,310 98 136 172 11.2 12.5 13.1 Let's suppose that, by measure of prudence, that this estimated coefficient of variation is rounded to 13%, and that one wishes to estimate the average weight in the future with a maximum relative mistake of 5% (that means a maximum mistake of 50 gr for an average weight of 1,000gr, of 60 gr for a middleweight of 1,200gr, etc.). One has in these conditions: 2 2 134 27 5 n = A result more exact 28n = or 29n = can be gotten with the help of the diagram, while joining by a right line the points 13%V = and 5%rd = , and while reading the corresponding value on the " "n axis. Inversely, one can wonder what precision one would get while weighting 50 chosen chickens at random in the considered population. One would have then, for a same degree of confidence level: 132 3.7% 50rd = This result means that one would only commit 5% of the cases an error superior to 37 gr for an average weight of 1,000gr, to 44 gr for a middleweight of 1,200gr, etc. 148 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 2.1.3. Approximation of the population Standard deviation under the normal law distribution In many cases the standard deviation of the population is not available. Since we have not yet taken the sample and are in the stage of deciding how large to make it (sample), we cannot estimate the population standard deviation. In such a situation, if we have an idea about the range ( i.e., the difference between the highest and the lowest values of the population), we can use that to get a crude estimate of the standard deviation of the population for getting a working idea of the required sample size. We can get the said estimate of standard deviation as follows: Since 99.73 per cent of the area under normal curve lies within the range 3± standard deviations, we may say that these limits include almost all of the distribution. Accordingly, we can say that the given range equals 6 standard deviations because of 3± . Thus, a rough estimate of the population standard deviation would be:  6 6 the given rangethe given rangeσ σ= ⇒ = If the range happens to be, say $12, then  12 $2 6 σ = = and this estimate of standard deviation σ can be used to determine the sample size in the formulae stated above 2.1.4. Case of Standard deviation and Coefficient of variation unknown When the standard deviation and the coefficient of variation are unknown or known pain, one can use a more rigorous method, presented notably by [4]. If one has an estimation  2 σ of the variance of the population, with 2k degrees of freedom, and if one sets an absolute error rd , corresponding to a confidence level 1 α− and as it is only passed with a probability β , the number of measures to be done is:  2 2 1 21 2 r n t F dα β σ − − = (2.8) The value 1 2 t α − possessing 1n − degrees of freedom and the value 1F β− of Snedecor possessing 1 1k n= − and 2k degrees of freedom. The most often, the introduction of the factor 1F β− provokes an increase of n of the order of 25 to 75%, in relation to the case where the coefficient of variation is supposed known. Example: The following tables give the estimated values of “n” 149 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 2.2: Approximate values of the sample size “n” needed for the error dr =5% while the population variance is estimated to 4 with k2 =30 degrees of freedom dr =5% α=0.05 α=0.01 β=0.05 β=0.01 β=0.05 β=0.01 1,077,822 1,954,037 27,038,918 49,020,196 98,356 159,680 523,262 849,513 47,305 73,063 159,396 246,190 33,167 49,566 91,231 136,338 26,757 39,131 65,808 96,242 23,185 33,244 53,208 76,295 20,852 29,532 45,642 64,643 19,314 26,971 40,882 57,091 18,092 25,051 37,349 51,714 17,156 23,668 34,707 47,883 16,432 22,478 32,723 44,763 15,877 21,575 31,210 42,409 15,018 20,171 28,927 38,853 14,310 19,128 27,167 36,313 13,437 17,754 24,994 33,024 12,883 16,836 23,657 30,917 12,276 15,878 22,264 28,798 11,709 14,980 20,940 26,790 11,366 14,465 20,195 25,703 10,644 13,415 18,646 23,501 10,132 12,541 17,548 21,721 9,957 12,355 17,200 21,341 The next table gives the approximate values of the sample size “n” needed for the error dr =10% while the population variance is estimated to 4 with k2 =30 degrees of freedom. We remark from the next tables that, as the error dr increases, as the needed sample size decreases. We also remark that the sample size for all cases is relatively high; that is why this approach is not practical. 150 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 2.2: Approximate values of the sample size “n” needed for the error dr =10% while the population variance is estimated to 4 with k2 =30 degrees of freedom. dr=10% α=0.05 α=0.01 β=0.05 β=0.01 β=0.05 β=0.01 269,456 488,509 6,759,729 12,255,049 24,589 39,920 130,815 212,378 11,826 18,266 39,849 61,548 8,292 12,392 22,808 34,084 6,689 9,783 16,452 24,060 5,796 8,311 13,302 19,074 5,213 7,383 11,410 16,161 4,828 6,743 10,220 14,273 4,523 6,263 9,337 12,929 4,289 5,917 8,677 11,971 4,108 5,620 8,181 11,191 3,969 5,394 7,802 10,602 3,754 5,043 7,232 9,713 3,578 4,782 6,792 9,078 3,359 4,438 6,249 8,256 3,221 4,209 5,914 7,729 3,069 3,970 5,566 7,200 2,927 3,745 5,235 6,697 2,841 3,616 5,049 6,426 2,661 3,354 4,662 5,875 2,533 3,135 4,387 5,430 2,489 3,089 4,300 5,335 2.2. Sampling Distribution of proportion (frequency) The variance of the sampling distribution of proportion or frequency f is given by ( )2 pqf n σ = for infinite population or ( )2 1 pq N nf n N σ − = − for an exhaustive sampling from a finite population of size N . Since the confidence interval for the universe proportion, p is given by 151 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 c pqf z n ± × then with the given precision rate, the acceptable error d can be expressed as follows: 2 2 2 2 c c c pq pqd z d z n n z p qn d = ⇔ = × × = (2.9) The formula gives the size of the sample in case of infinite population when we are to estimate the proportion in the universe. But in case of finite population the above stated formula becomes [4] ( ) ( ) 2 2 21 c c z p q Nn N d z p q × × × = − × + × × (2.10) where p represents the population reliability and 1q p= − . Notice that for a no reliable population we take 0.5p = 3. Determination of sample size trough the approach based on Bayesian statistics This approach of determining “n” utilizes Bayesian statistics and as such is known as Bayesian approach. The procedure for finding the optimal value of “n” or the size of sample under this approach is as follows [4] expected value of the sample information (EVSI) for every possible n; (i) Also workout reasonably approximated cost of taking a sample for every possible n; (ii) Compare the EVSI and the cost of the sample for every possible n. In other words, workout the expected net gain (ENG) for every possible n as stated below: For a given sample size (n): (EVSI) – (Cost of sample) = (ENG) (2.11) Form (iii) above the optimal sample size, that value of n which maximizes the difference between the EVSI and the cost of the sample, can be determined. The computation for every possible n and then comparing the same with the respective cost is often a very cumbersome task and is generally feasible with mechanized or computer help. Hence, this approach although being theoretically optimal is rarely used in practice. 4. Results The following are the examples of determination of sample size knowing the error d and the population probability (reliability) p , for different population size N using the formula (2.10). the size would be rounded to the nearest positive integer. 152 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 4.1: Sample size n from a population with N individuals and error d=1% p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 100 98.98 99.41 98.94 99.38 98.79 99.30 98.41 99.08 97.22 98.37 120 118.53 119.15 118.47 119.11 118.26 118.99 117.72 118.67 116.01 117.66 150 147.71 148.67 147.61 148.61 147.28 148.42 146.45 147.93 143.80 146.36 200 195.94 197.64 195.77 197.54 195.19 197.19 193.73 196.33 189.12 193.57 250 243.68 246.31 243.43 246.16 242.51 245.62 240.27 244.29 233.20 240.02 300 290.94 294.70 290.58 294.49 289.28 293.72 286.08 291.81 276.12 285.74 400 384.04 390.63 383.41 390.25 381.15 388.90 375.62 385.56 358.61 375.02 500 475.30 485.44 474.33 484.86 470.87 482.77 462.46 477.62 436.94 461.55 700 652.51 671.78 650.67 670.66 644.18 666.66 628.52 656.89 582.28 626.86 1000 905.78 943.37 902.24 941.15 889.81 933.30 860.19 914.24 775.83 857.08 1500 1297.49 1376.05 1290.23 1371.33 1264.96 1354.72 1205.91 1314.93 1046.35 1199.79 2000 1655.43 1785.52 1643.64 1777.57 1602.84 1749.77 1509.18 1683.93 1267.29 1499.61 2500 1983.81 2173.59 1966.88 2161.83 1908.74 2120.85 1777.37 2024.88 1451.14 1764.11 3000 2286.12 2541.90 2263.68 2525.83 2187.00 2470.06 2016.24 2340.84 1606.51 1999.19 5000 3288.37 3844.96 3242.12 3808.31 3087.07 3682.91 2757.40 3402.80 2044.26 2725.61 7500 4211.54 5170.15 4135.98 5104.09 3886.92 4881.32 3378.33 4401.11 2366.72 3330.73 10000 4899.25 6246.62 4797.29 6150.43 4465.40 5829.83 3806.97 5157.69 2569.36 3746.62 20000 6488.53 9083.52 6310.89 8881.53 5748.78 8228.07 4701.80 6949.76 2947.97 4610.09 50000 8056.64 12485.74 7784.55 12107.25 6946.67 10924.51 5473.77 8780.19 3233.89 5349.87 100000 8762.53 14266.98 8441.62 13774.91 7465.19 12264.22 5790.69 9625.23 3341.93 5652.21 200000 9163.99 15362.81 8813.58 14793.76 7754.60 13065.34 5963.32 10111.82 3398.70 5816.56 500000 9423.02 16105.02 9052.93 15480.77 7939.28 13598.30 6071.93 10428.14 3433.70 5919.84 1000000 9512.65 16368.63 9135.62 15724.18 8002.81 13785.75 6109.02 10538.02 3445.53 5955.09 10000000 9594.79 16613.36 9211.35 15949.88 8060.86 13958.93 6142.78 10638.91 3456.25 5987.17 10000000 9603.08 16638.23 9218.99 15972.81 8066.71 13976.49 6146.18 10649.11 3457.32 5990.40 153 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 4.2: Sample size n from a population with N individuals and error d=2 % p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 96.04 97.68 95.88 97.58 95.32 97.25 93.95 96.42 89.72 93.80 120 114.33 116.66 114.11 116.53 113.31 116.05 111.37 114.87 105.48 111.17 150 141.24 144.81 140.89 144.61 139.68 143.87 136.74 142.05 127.94 136.43 200 184.69 190.87 184.11 190.51 182.04 189.22 177.07 186.09 162.57 176.54 250 226.51 235.88 225.63 235.33 222.53 233.37 215.14 228.62 194.09 214.36 300 266.78 279.88 265.55 279.10 261.27 276.35 251.13 269.71 222.90 250.07 400 343.00 364.99 340.98 363.67 333.94 359.01 317.55 347.87 273.67 315.85 500 413.97 446.45 411.02 444.47 400.83 437.53 377.43 421.08 317.00 375.04 700 542.16 599.31 537.12 595.73 519.83 583.32 481.14 554.44 387.02 477.26 1000 706.18 806.37 697.64 799.91 668.75 777.68 606.02 727.17 463.87 599.87 1500 923.46 1102.69 908.91 1090.65 860.47 1049.72 759.30 959.70 548.60 749.67 2000 1091.36 1350.89 1071.09 1332.87 1004.44 1272.25 869.23 1142.35 603.74 856.63 2500 1225.00 1561.83 1199.51 1537.79 1116.54 1457.64 951.92 1289.61 642.48 936.83 3000 1333.89 1743.30 1303.72 1713.40 1206.28 1614.48 1016.38 1410.86 671.20 999.19 5000 1622.30 2271.06 1577.88 2220.57 1437.35 2057.20 1175.58 1737.61 737.09 1152.66 7500 1818.94 2676.15 1763.29 2606.30 1589.59 2384.07 1275.48 1965.17 775.13 1248.53 10000 1936.29 2938.19 1873.35 2854.20 1678.48 2589.83 1332.08 2102.87 795.67 1302.71 20000 2143.75 3444.02 2066.86 3329.19 1832.17 2974.95 1427.07 2349.85 828.59 1393.41 50000 2291.03 3840.76 2203.43 3698.49 1938.68 3266.38 1490.85 2527.99 849.69 1454.16 100000 2344.73 3994.12 2253.05 3840.50 1976.99 3376.64 1513.40 2593.53 856.96 1475.60 200000 2372.53 4075.49 2278.71 3915.67 1996.71 3434.61 1524.93 2627.59 860.64 1486.57 500000 2389.53 4125.93 2294.39 3962.20 2008.74 3470.36 1531.93 2648.46 862.87 1493.22 1000000 2395.25 4143.02 2299.66 3977.96 2012.78 3482.44 1534.28 2655.49 863.61 1495.45 10000000 2400.42 4158.52 2304.43 3992.25 2016.43 3493.39 1536.40 2661.85 864.29 1497.47 154 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 Table 4.3: Sample size n from a population with N individuals and error d=3 % p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 91.51 94.92 91.19 94.72 90.05 94.01 87.34 92.28 79.51 87.05 120 107.96 112.74 107.51 112.46 105.94 111.46 102.19 109.04 91.62 101.80 150 131.62 138.81 130.95 138.38 128.62 136.87 123.14 133.23 108.08 122.56 200 168.57 180.57 167.47 179.84 163.67 177.29 154.87 171.21 131.75 153.97 250 202.70 220.33 201.12 219.24 195.65 215.46 183.20 206.54 151.68 181.94 300 234.34 258.24 232.22 256.75 224.96 251.57 208.65 239.49 168.70 207.01 400 291.14 329.00 287.88 326.59 276.79 318.24 252.49 299.14 196.21 250.09 500 340.69 393.74 336.22 390.28 321.19 378.42 288.91 351.70 217.49 285.77 700 422.95 507.97 416.09 502.23 393.30 482.74 345.94 440.06 248.27 341.44 1,000 516.48 649.23 506.28 639.88 472.93 608.57 406.05 542.24 277.74 399.87 1,500 623.77 828.41 608.95 813.23 561.32 763.31 469.50 661.75 306.00 461.26 2,000 696.07 961.02 677.66 940.66 619.18 874.49 509.29 743.70 322.40 499.61 2,500 748.09 1,063.13 726.87 1,038.27 660.00 958.23 536.58 803.40 333.11 525.84 3,000 787.32 1,144.18 763.85 1,115.43 690.34 1,023.58 556.46 848.82 340.65 544.92 5,000 879.57 1,350.03 850.37 1,310.18 760.23 1,185.23 600.98 957.05 356.82 587.54 7,500 934.30 1,483.47 901.42 1,435.49 800.77 1,286.84 626.03 1,022.21 365.49 611.45 10,000 964.31 1,560.60 929.32 1,507.59 822.71 1,344.48 639.35 1,058.24 369.98 624.16 20,000 1,013.11 1,692.60 974.56 1,630.42 857.96 1,441.30 660.43 1,117.31 376.94 644.23 50,000 1,044.83 1,783.10 1,003.88 1,714.22 880.60 1,506.40 673.76 1,156.02 381.24 656.91 155 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 100,000 1,055.85 1,815.45 1,014.05 1,744.10 888.42 1,529.42 678.33 1,169.53 382.69 661.25 200,000 1,061.45 1,832.07 1,019.21 1,759.43 892.38 1,541.20 680.63 1,176.41 383.43 663.44 500,000 1,064.84 1,842.19 1,022.33 1,768.76 894.77 1,548.35 682.02 1,180.57 383.87 664.76 1,000,000 1,065.97 1,845.59 1,023.38 1,771.90 895.57 1,550.75 682.49 1,181.96 384.01 665.20 10,000,000 1,067.00 1,848.66 1,024.32 1,774.73 896.29 1,552.92 682.90 1,183.22 384.15 665.60 Table 4.4: Sample size n from a population with N individuals and error d=4 % p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 85.84 91.31 85.34 90.98 83.59 89.82 79.51 87.05 68.58 79.09 120 100.15 107.68 99.46 107.22 97.09 105.61 91.62 101.80 77.38 91.06 150 120.17 131.20 119.18 130.52 115.78 128.15 108.08 122.56 88.78 107.30 200 150.20 167.88 148.66 166.76 143.40 162.90 131.75 153.97 104.12 130.59 250 176.70 201.71 174.57 200.10 167.35 194.55 151.68 181.94 116.15 150.15 300 200.25 233.01 197.51 230.86 188.32 223.51 168.70 207.01 125.86 166.80 400 240.28 289.09 236.35 285.79 223.30 274.59 196.21 250.09 140.53 193.64 500 273.03 337.89 267.96 333.38 251.30 318.24 217.49 285.77 151.09 214.34 700 323.40 418.64 316.31 411.75 293.34 388.87 248.27 341.44 165.30 244.17 1,000 375.33 510.07 365.81 499.86 335.42 466.53 277.74 399.87 177.84 272.62 1,500 428.90 614.44 416.51 599.69 377.55 552.33 306.00 461.26 188.99 299.79 2,000 461.86 684.46 447.52 666.20 402.85 608.25 322.40 499.61 195.11 315.51 156 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 2,500 484.19 734.70 468.45 713.70 419.73 647.60 333.11 525.84 198.97 325.76 3,000 500.31 772.50 483.53 749.32 431.78 676.79 340.65 544.92 201.63 332.98 5,000 536.01 861.11 516.78 832.40 458.11 743.83 356.82 587.54 207.18 348.40 7,500 555.84 913.50 535.19 881.26 472.51 782.59 365.49 611.45 210.07 356.66 10,000 566.31 942.17 544.90 907.90 480.05 803.53 369.98 624.16 211.54 360.94 20,000 582.79 988.70 560.13 951.03 491.84 837.13 376.94 644.23 213.79 367.56 50,000 593.14 1,018.89 569.69 978.93 499.19 858.67 381.24 656.91 215.16 371.65 100,000 596.67 1,029.37 572.94 988.60 501.69 866.09 382.69 661.25 215.63 373.03 200,000 598.46 1,034.69 574.59 993.51 502.94 869.86 383.43 663.44 215.86 373.72 500,000 599.53 1,037.91 575.58 996.47 503.70 872.13 383.87 664.76 216.00 374.14 1,000,000 599.89 1,038.98 575.91 997.47 503.96 872.89 384.01 665.20 216.04 374.28 10,000,000 600.21 1,039.95 576.21 998.36 504.18 873.58 384.15 665.60 216.09 374.41 Table 4.5: Sample size n from a population with N individuals and error d=5% p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 79.51 87.05 78.84 86.59 76.52 84.96 71.29 81.14 58.28 70.76 120 91.62 101.80 90.73 101.16 87.67 98.94 80.86 93.80 64.50 80.18 150 108.08 122.56 106.84 121.64 102.62 118.44 93.40 111.13 72.21 92.49 200 131.75 153.97 129.90 152.51 123.71 147.50 110.53 136.32 82.00 109.26 250 151.68 181.94 149.24 179.90 141.11 172.97 124.21 157.78 89.27 122.60 300 168.70 207.01 165.68 204.37 155.72 195.47 135.37 176.28 94.88 133.47 400 196.21 250.09 192.13 246.24 178.85 233.43 152.51 206.55 102.96 150.09 500 217.49 285.77 212.49 280.76 196.36 264.21 165.04 230.27 108.50 162.21 157 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 700 248.27 341.44 241.77 334.31 221.09 311.09 182.15 265.07 115.62 178.71 1,000 277.74 399.87 269.63 390.12 244.15 358.85 197.50 298.95 121.60 193.46 1,500 306.00 461.26 296.17 448.32 265.71 407.51 211.36 331.95 126.70 206.74 2,000 322.40 499.61 311.51 484.47 277.98 437.14 219.04 351.35 129.41 214.09 2,500 333.11 525.84 321.50 509.09 285.90 457.09 223.93 364.11 131.10 218.75 3,000 340.65 544.92 328.52 526.95 291.44 471.43 227.31 373.15 132.25 221.97 5,000 356.82 587.54 343.52 566.70 303.19 502.99 234.38 392.63 134.60 228.71 7,500 365.49 611.45 351.55 588.92 309.42 520.41 238.09 403.16 135.81 232.24 10,000 369.98 624.16 355.71 600.69 312.64 529.58 239.99 408.64 136.42 234.05 20,000 376.94 644.23 362.13 619.26 317.59 543.96 242.89 417.15 137.35 236.80 50,000 381.24 656.91 366.10 630.96 320.63 552.96 244.66 422.42 137.92 238.49 100,000 382.69 661.25 367.44 634.96 321.66 556.03 245.26 424.21 138.11 239.06 200,000 383.43 663.44 368.12 636.98 322.18 557.58 245.56 425.11 138.20 239.34 500,000 383.87 664.76 368.52 638.20 322.49 558.51 245.74 425.65 138.26 239.52 1,000,000 384.01 665.20 368.66 638.61 322.59 558.83 245.80 425.83 138.28 239.57 10,000,000 384.15 665.60 368.78 638.97 322.68 559.11 245.86 425.99 138.30 239.62 Table 4.6: Sample size n from a population with N individuals and error d=6% p=0.5 p=0.6 p=0.7 p=0.8 p=0.9 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 72.93 82.36 72.12 81.76 69.36 79.68 63.30 74.93 49.24 62.70 120 82.98 95.43 81.93 94.63 78.38 91.85 70.71 85.58 53.59 69.97 150 96.25 113.44 94.83 112.30 90.10 108.40 80.10 99.76 58.79 79.14 200 114.55 139.81 112.55 138.08 105.93 132.23 92.36 119.57 65.10 91.08 250 129.31 162.48 126.76 160.14 118.42 152.32 101.69 135.75 69.59 100.15 300 141.46 182.17 138.41 179.23 128.52 169.49 109.04 149.20 72.93 107.27 400 160.28 214.69 156.38 210.62 143.86 197.28 119.87 170.31 77.60 117.73 500 174.19 240.44 169.58 235.35 154.95 218.81 127.47 186.10 80.70 125.04 700 193.36 278.64 187.70 271.83 169.93 249.98 137.42 208.16 84.56 134.60 158 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 1000 210.76 316.34 204.05 307.58 183.22 279.89 145.96 228.48 87.70 142.79 1500 226.62 353.54 218.88 342.63 195.08 308.61 153.38 247.24 90.32 149.88 2000 235.48 375.62 227.13 363.33 201.60 325.30 157.38 257.83 91.68 153.70 2500 241.14 390.25 232.39 376.99 205.73 336.21 159.88 264.63 92.52 156.08 3000 245.07 400.65 236.04 386.69 208.58 343.89 161.59 269.37 93.09 157.71 5000 253.31 423.21 243.67 407.66 214.52 360.38 165.13 279.37 94.25 161.08 7500 257.65 435.47 247.68 419.02 217.62 369.22 166.96 284.65 94.84 162.82 10000 259.87 441.87 249.74 424.95 219.20 373.81 167.89 287.37 95.14 163.70 20000 263.28 451.83 252.88 434.15 221.62 380.91 169.30 291.54 95.59 165.04 50000 265.37 458.02 254.81 439.86 223.10 385.31 170.16 294.11 95.86 165.86 100000 266.07 460.13 255.45 441.80 223.59 386.79 170.45 294.97 95.95 166.14 200000 266.42 461.19 255.78 442.78 223.84 387.54 170.59 295.40 95.99 166.27 500000 266.64 461.82 255.98 443.37 223.99 387.99 170.68 295.67 96.02 166.35 1000000 266.71 462.04 256.04 443.56 224.04 388.14 170.71 295.75 96.03 166.38 10000000 266.77 462.23 256.10 443.74 224.09 388.27 170.73 295.83 96.04 166.41 Table 4.7: Sample size n from a population with N individuals and error d=10% p=0.5 p=0.4 p=0.3 p=0.2 p=0.1 α 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 N 100 49.24 62.70 48.22 61.74 44.90 58.54 38.30 51.83 25.88 37.70 120 53.59 69.97 52.39 68.77 48.48 64.82 40.87 56.67 27.02 40.18 150 58.79 79.14 57.34 77.61 52.69 72.61 43.81 62.53 28.25 43.01 200 65.10 91.08 63.32 89.06 57.69 82.52 47.20 69.72 29.60 46.28 250 69.59 100.15 67.55 97.71 61.18 89.89 49.49 74.90 30.48 48.48 300 72.93 107.27 70.70 104.47 63.74 95.57 51.16 78.79 31.09 50.07 400 77.60 117.73 75.08 114.36 67.27 103.78 53.39 84.27 31.90 52.22 500 80.70 125.04 77.98 121.25 69.59 109.41 54.83 87.95 32.40 53.59 700 84.56 134.60 81.57 130.22 72.43 116.66 56.58 92.55 32.99 55.26 159 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 1,000 87.70 142.79 84.49 137.87 74.72 122.75 57.96 96.34 33.45 56.57 1,500 90.32 149.88 86.91 144.46 76.60 127.95 59.08 99.50 33.82 57.64 2,000 91.68 153.70 88.18 148.01 77.58 130.71 59.66 101.17 34.00 58.19 2,500 92.52 156.08 88.95 150.21 78.18 132.43 60.01 102.19 34.12 58.53 3,000 93.09 157.71 89.48 151.72 78.59 133.60 60.25 102.88 34.19 58.75 5,000 94.25 161.08 90.55 154.84 79.41 136.01 60.73 104.30 34.34 59.21 7,500 94.84 162.82 91.09 156.44 79.83 137.24 60.97 105.03 34.42 59.44 10,000 95.14 163.70 91.37 157.26 80.04 137.87 61.10 105.39 34.46 59.56 20,000 95.59 165.04 91.78 158.50 80.35 138.82 61.28 105.94 34.52 59.73 50,000 95.86 165.86 92.03 159.25 80.55 139.40 61.39 106.28 34.55 59.84 100,000 95.95 166.14 92.11 159.50 80.61 139.59 61.43 106.39 34.56 59.87 200,000 95.99 166.27 92.16 159.63 80.64 139.69 61.45 106.45 34.57 59.89 500,000 96.02 166.35 92.18 159.70 80.66 139.75 61.46 106.48 34.57 59.90 1,000,000 96.03 166.38 92.19 159.73 80.67 139.77 61.46 106.49 34.57 59.90 10,000,000 96.04 166.41 92.20 159.75 80.67 139.78 61.47 106.50 34.57 59.91 5. Concluding remarks The size of the sample affects the accuracy of the inference made about the population. The larger the sample, the closer one is to measuring the population itself. Therefore, large samples are more useful than small samples, if the large samples are unbiased, i.e., random. On the other hand, a small unbiased sample is certainly more useful than a large biased sample. Therefore, the single most important characteristic of a good sample is not its size but its representative nature, i.e., randomness. It must be realized that the time and expense involved in increasing sample sizes beyond certain limits may result in diminishing returns. The tables above show that for the sampling distribution of the frequency, the necessary sample size to achieve the desired accuracy depends upon the reliability of the population; the bigger the population reliability, the smaller is the sample size required. References [1] Dagnelie P., Principes d’experimentation. Gembloux. Belgium. 2001, pp 88-92. [2] Gurumani N., An introduction to Biostatistics. 2nd Edition. MJP Publishers. India, 2005, pp. 185-186. 160 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 12, No 1, pp 143-161 [3] Harris H.M, Horvitz D.G., Mood A.M. On the determination of sample sizes in designing experiments. J. Amer. Statist. 35. USA. 1948. Pp. 391-402 [4] Kothari. C.R., Research Methodology. Methods and Techniques. New Age International (P) Ltd., Publishers. Dharmesh Art Process, Delhi. India, 2008, pp. 174-180 [5] Lenth R.V., Some practical guidelines for effective sample determination. Am. Statist. 55(3), 2001. Pp. 187- 193 [6] Roger C. Pfaffenberger, James H. Patterson, Statistical methods. Third edition. IRWIN, Homewood. Illinois. USA, 1987, pp. 387-392 161