在統計的應用上，想要對母體平均數 做區間估計及檢定時，需要用到樣本平均數 之抽樣分布，通常需要依賴中央極限定理來得到。然而中央極限定理是一個極限結果，在實際應用時必須樣本數 n「足夠大」才適用，怎樣是「足夠大」卻無定論。 怎樣的 n 才適用中央極限定理，其中以 n 大於或等於30似被較多教科書採用。然而對於很偏的 (skewed) 分布來說， n 等於30或40時若應用中央極限定理，會不會得出一些誤導的結論？這是我們在本文所探討的問題。 本論文將利用電腦模擬方式，探討在不對稱分布情況下，隨著樣本數 n 的變動，應用中央極限定理的結果會如何。所討論的分布以 gamma 分布、混合常態的雙峰 (bimodal) 分布及伯努利 (Bernoulli) 分布為主。討論的結果發現，當我們用樣本標準差 S 代替母體標準差 並應用中央極限定理計算信賴區間及處理檢定問題時，信賴係數及顯著水準 並不如我們的預期。另外我們在討論中央極限定理應用於離散型的二項分布時發現，當 p 值很小或很大的情況下，即使滿足書上建議 np 與 n(1-p) 同時大於或等於5甚至10時，在估計 p 的信賴區間之信賴係數時，也有一些問題必須注意。 When we make statistical inferences about the population mean based on the sample mean , we often rely on the central limit theorem to obtain the （approximate）sampling distribution of . The central limit theorem is an asymptotic result, hence the sample size has to be sufficiently large for the application to be appropriate. Yet there does not seem to be much discussion on how large qualifies for “sufficiently large”. General suggestions have been made which include, for example, , or . But we do know that if the population distribution is very skewed, then it takes bigger sample size for the distribution of the sample mean to be close to normal distribution. We would like to explore in more details about the appropriateness of the application of the central limit theorem under these circumstances. We used computer to simulate random samples from gamma, mixed normal and Bernoulli distributions and found that when the population standard deviation is unknown and has to be replaced by the sample standard deviation, cautions have to be taken when one applies the central limit theorem and makes decisions based on the result of the inferences, because the significant levels and confidence coefficients may not be what we expected.
