在中央極限定理（Central Limit Theorem）的基礎上，當樣本數大於30 時，其樣本平均數的抽樣分配是近似常態分配，也就是在中央極限定理下，常態分配可當做不少大樣本的近似分配。然在現實生活中存在著各式形態的機率分配；如有單峰、多峰之分配；對稱、不對稱之分配；高、低偏態之分配；以及無偏、無峰、無尾且對稱之均勻分配，各個機率分配之形態有的與常態分配類似，有的則差距很大，甚至有由常態分配所衍生的其他分配，如 t分配、卡方分配、F 分配，其中 t分配形如標準常態分配，卡方分配、F 分配形如Gamma分配，其峰度及偏態隨自由度而變化。本文欲探究樣本數(自由度)應該要多大，t分配、卡方分配及二項分配才可以被接受近似常態分配，而以常態分配所取代，也探討在實際應用中央極限定理時，一般教科書所建議的樣本大小是否合適的問題。
本研究試圖運用電腦查表方式，針對 t分配、卡方分配與標準常態分配臨界值的誤差做比較及使用電腦模擬方式，針對卡方分配及二項分配樣本平均數的抽樣分配近似常態分配所需的最小樣本數(自由度)做探討，並在各章節提供各機率分配之近似常態分配所需的最小樣本數(自由度)之參考表格。

According to the Central Limit Theorem, when the sample n>30 size  , the sampling distribution of sample mean   will be approximated to the normal distribution. In other words, the normal distribution can be used as the approximate distribution of many types of samples under the Central Limit Theorem. In real life, there are assorted types of probability distribution, such as the unimodal vs multimodal distributions, the symmetrical vs asymmetrical distributions, the high vs low skewness distributions, and the non-skewed, nonmodal, and no-tail uniform distribution. While some of probability distributions have a normal distribution-like pattern, others may have a pattern that differs greatly from the normal distribution pattern. Take the normal distribution derived t-distribution,   chi-square distribution, F distribution as an example, even though the t-distribution has a shape like that of the standard normal distribution, and chi-square distribution, such as the gamma distribution, its kurtosis and skewness change according to the degree of freedom, commonly. The purpose of this thesis is to explore the minimum sample size ( degrees of freedom) required of the t-distribution, chi-square distribution and binomial distribution for the normal approximation and be replaced by the normal distribution and to explore the investigators examined the appropriateness of using the sample size suggested by general textbooks for determining whether the Central limit theorem can be used or not. 
It was done using the computer technology to compare the critical value of the error at the t-distribution,chi-square distribution and Z-distribution and used the computer simulation to explore the minimum sample size ( degrees of freedom) and required for the means of the chi-square distribution and binomial distribution to be approximated to the normal distribution The minimum sample size (degrees of freedom) required for the normal distribution approximating to each probability distribution is also listed in the table in various chapters for reference.

目 錄
表目錄…Ⅲ
圖目錄…Ⅳ
使用符號一覽表…Ⅴ 
                             
第一章  緒 論…1
1.1  研究動機與目的…1
1.2  文獻探討…2
1.3  研究方法…5
1.4  研究架構…6

第二章  T分配近似常態分配速度之探討…8
2.1  t分配近似標準常態分配之檢驗…8
2.2  t分配臨界值近似標準常態分配臨界值之速度…15
2.3  討 論…27

第三章   分配近似常態分配速度之探討…28
3.1  中央極限定理於 分配上之樣本數探討…28
3.2  分配標準常態化臨界值近似標準常態分配臨界值之速度…35
3.3  討 論…45

第四章  二項分配近似常態分配速度之探討…46
4.1  模 擬…46
4.2  分 析…47
4.3  討 論…55

第五章  結 論…61
5.1  研究結論…61
5.2  未來研究方向…63

參考文獻…64

表 目 錄
表2.1  t分配在 {0.1587, 0.0228, 0.00135}之臨界值…11
表2.2  t分配與Z分配在 {0.1587, 0.0228, 0.00135}下之誤差值…13
表2.3  t分配在 {0.1, 0.05, 0.025, 0.01, 0.005}之臨界值…18
表2.4  t分配與Z分配在 {0.1, 0.05, 0.025, 0.01, 0.005}下之誤差值…22
表2.5  t分配近似標準常態分配所需之最小自由度…27
表3.1  卡方分配之W檢定結果…31
表3.2  卡方分配在 {0.1, 0.05, 0.025, 0.01, 0.005}標準常態化值…38
表3.3  卡方分配標準常態化與Z分配之誤差值…41
表3.4  卡方分配近似常態分配所需之最小自由度…44
表3.5  卡方分配之迴歸模式的判定係數…44
表4.1  二項分配之W檢定結果…49
表4.2  二項分配之倒數迴歸模式…56
表4.3  二項分配在不同要求拒絕率下之迴歸模式…59
表4.4  二項分配應用中央極限定理所需之最小樣本數…60

圖 目 錄
圖1.1  研究架構…7
圖2.1  在常態分配曲線下之面積…8
圖2.2  在標準常態分配曲線下之面積…9
圖2.3  顯示同時誤差值0～0.2其t分配所需最小自由度…10
圖2.4  標準常態分配與t分配曲線的比較…16
圖2.5  t分配的機率密度函數圖形…17
圖2.6  顯示誤差值0.01～0.05其t分配所需最小自由度…26
圖3.1  卡方分配的機率密度函數圖形…29
圖3.2  卡方分配其自由度 與拒絕常態性檢定次數比例之關係…34
圖3.3  顯示誤差值0.04～0.08其卡方分配所需最小自由度…37
圖4.1  二項分配其樣本數 與拒絕常態性檢定次數之關係…55
圖4.2  二項分配之倒數迴歸模式曲線…57
圖4.3  二項分配在不同要求拒絕率 下之伯努力試驗結果示意…58
圖4.4  二項分配之參數與樣本數之迴歸模式曲線…59

1.	Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2014), Statistics for business and economics (12th ed.). Boston: Cengage Learning.
2.	Andersson, M. K. (2001), A normality test for the mean estimator. Applied Economics Letters, 8(9), 633-634.
3.	Atukorala, R., King, M. L., & Sriananthakumar, S. (2015), Applications of information measures to assess convergence in the central limit theorem. Model Assisted Statistics and Applications, 10(3), 265-276. 
4.	Bernoulli, J. (2005), On the law of large numbers. Berlin: Oscar Sheynin. 
5.	Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2014), Basic business statistics: concepts and applications (13rd ed.). New York: Pearson Ed.
6.	Berry, D. A., & Lindgren, B. W. (1995), Statistics: theory and methods(2nd ed.).Plymouth: Duxbury Press.
7.	Box, G. E. P., Hunter, W. G., & Hunter, J. S. (2005), Statistics for experiments: an introduction to design, data analysis and modeling (2nd ed.). New York: Wiley. 
8.	Chang, H. J. (2002), Statistics. Taipei: Hwa Tai.	
9.	Chang, H. J., Huang, K. C., & Wu, C. H. (2006), Determination of sample size in using central limit theorem for weibull distribution. International Journal of Information and Management Sciences, 17(3), 31-46.
10.	Chang, H. J., Wu, C. H., Ho, J. F., & Chen, P. Y. (2008), On sample size in using central limit theorem for gamma distribution. International Journal of Information and Management Sciences, 19(1), 153-174.
11.	Cortinhas, C., & Black, K. (2014), Statistics for business and economics. Hoboken: Wiley.
12.	Devore, J. L. (2011), Probability and statistics for engineering and the sciences (8th ed.). Boston: Brooks/Cole.  
13.	Devore, J. L., & Berk, K. N. (2007). Modern mathematical statistics with applications. 
   New York: Wiley.
14.	De Moivre, A. (2013). The doctrine of chances. New York: Cambridge Univ. 
15.	Fisher, R. A. (1925). Statistical methods for research workers. New Delhi: Cosmo.
16.	Fitrianto, A., & Hanafi, I. (2014). Exploring central limit theorem on world population density data. AIP Conference Proceedings , 1635(1), 737-741.  
17.	Gosset ,W.S. (1908). The probable error of a mean. Biometrika, 6(1), 1-25.
18.	Hodges Jr, J. L., & Lehmann, E. L. (2005). Basic concepts of probability and statistics (2nd ed.). Philadelphia: Society for Industrial and Applied Math.
19.	Hoeffding, W., & Robbins, H. (1948). The central limit theorem for dependent random variables. Duke math. J, 15(3), 773-780.
20.	Hogg, R. V., Tanis, E. A., & Rao, M. J. M. (2010). Probability and statistical inference. 
NY: Macmillan.
21.	Jolliffe, I. T. (1995). Sample sizes and the central limit theorem: the Poisson distribution as an illustration. The American Statistician, 49(3), 269-269.
22.	Keller, G. & Warrack, B. (2002). Statistics for management and economics (6th ed.). Pacific Grove: Thomson.
23.	Kim, H., Lovell, D. J., Kang, Y. S., & Kim, W. (2007). Data quantity for reliable traffic Information in vehicular ad hoc networks. In Vehicular Technology Conference, 2007. VTC-2007 Fall. 2007 IEEE (66 th), 1401-1405.
24.	Le Cam, L. (1986). The central limit theorem around 1935. Statistical Science, 78-91.
25.	Levine, D. M., Krehbiel, T. C., & Berenson, M. L. (2013). Business statistics: international ed., New York: Pearson Ed.
26.	Martínez–Abraín, A. (2014). Is the ‘N=30 rule of thumb’of ecological field studies reliable? A call for greater attention to the variability in our data. Animal Biodiversity and Conservation, 37(1), 95-100. 
27.	Olive, D. J. (2014). Bayesian methods. in statistical theory and inference. New York: Springer Pub.
28.	Pearson, E. S., D'agostino, R. B., & Bowman, K. O. (1977). Tests for departure from normality: comparison of powers. Biometrika, 231-246.
29.	Plackett, R. L. (1983). Karl pearson and the chi-squared test. International Statistical Review, 59-72.
30.	Plane, D. R., & Gordon, K. R. (1982). A simple proof of the nonapplicability of the central limit theorem to finite populations. The American Statistician, 36(3a), 175-176.
31.	Ramachandran, K. M., & Tsokos, C. P. (2014). Mathematical statistics with applications. Burlington: Academic Press.
32.	Royston, J. P. (1982). An extension of shapiro and wilk's w test for normality to large samples. J. Applied Statistics, 31(20), 115-124.
33.	Sanders, M. (2009), Characteristic function of the central chi-squared distribution.
 http://www.planetmathematics.com/CentralChiDistr.pdf, Last visited at 06/15/2017.
34.	Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3-4), 591-611.
35.	Shapiro, S. S., Wilk, M. B., & Chen, H. J. (1968). A comparative study of various tests for normality. Journal of the American Statistical Association, 63(324), 1343-1372.
36.	Shapiro, S. S., & Wilk, M. B. (1972). An analysis of variance test for the exponential distribution (complete samples). Technometrics, 14(2), 355-370.
37.	Walpole, R. E. (2011). Probability and statistics for engineering and the scientists (9th ed.). New York: Pearson.
38.	Walpole, R. E., Myers, R., Myers, S. L., & Keying, E. Y. (2012). Essentials of probabilty and statistics for engineers and scientists. New York: Pearson Ed.