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中文論文名稱 中央極限定理的樣本數模擬探討
英文論文名稱 ON SAMPLE SIZE SIMULATION IN CENTRAL LIMIT THEOREM
校院名稱 淡江大學
系所名稱(中) 管理科學研究所博士班
系所名稱(英) Graduate Institute of Management Science
學年度 93
學期 2
出版年 94
研究生中文姓名 吳昭賢
研究生英文姓名 Chao-Hsien Wu
學號 887560034
學位類別 博士
語文別 中文
口試日期 2005-06-17
論文頁數 91頁
口試委員 指導教授-張紘炬
委員-顏錫銘
委員-張逸民
委員-黃建森
委員-賴奎魁
委員-李培齊
委員-婁國仁
中文關鍵字 樣本數  亂數  模擬  Shapiro-Wilk  常態性檢定  隨機回答 
英文關鍵字 Sample Mean  Random Number  Simulation  Shapiro-Wilk  Normality Test  Randomized Response 
學科別分類
中文摘要 在許多統計的應用上,中央極限定理是非常重要的,只要有足夠大的樣本數,不需要去了解母體的分配型態甚至是已知母體並非是常態分配,中央極限定理就可以讓我們去推論母體的平均數。基於中央極限定理當樣本數足夠大時,樣本平均數的分配會近似常態分配。但是,要大到多少才叫足夠呢?在實際應用上,一般教科書或應用性之文獻或研究人員可能會使用樣本數是否大於30作為準則來假設樣本平均數的分配為常態分配。可是,現實上存在著各種型態的機率分配,各分配與常態分配之型態有類似,也有如天壤之別,因此,樣本數應該要多大才足夠來應用中央極限定理是本論文探討的主要目的。
本論文以電腦模擬的方式來找尋出各機率分配之抽樣樣本平均數近似常態分配所需之最小樣本數及其性質。在第二~五章中,以四個連續型分配為對象,包括均勻分配、三角分配、伽瑪分配與韋伯分配,探討中央極限定理在應用上所需之最小樣本數與其性質,並在各章節提供各機率分配之近似迴歸曲線與最小要求樣本數之參考表格,可供研究人員在實務操作上之參考。
最後,在第六章提出一對稱型之隨機回答調查設計,在研究敏感性問題時,具有下列優點:受訪者僅需要回覆一正整數或負整數,不需經過計算,任何人皆可輕易地勝任,且回答之過程不會為訪問者所查知,受訪者的隱私將可獲得良好的保障,受訪者進而更樂意去配合訪問與誠實作答。此隨機回答調查之設計相較於Christofides (2003)之設計,更加簡便可行。在6.2節中,也考慮在應用上述隨機回答調查設計時,樣本數之選擇問題,文中以三組有限離散型分配為例,探討其在應用中央極限定理所需之最小樣本數與其性質。
英文摘要 The central limit theorem is of crucial importance in many statistical applications. Given a large enough sample size, it enables us to make inferences about the population mean in cases where we do not know the specific shape of the population distribution and even in cases where we know that the population is not normally distributed. Based on the central limit theorem, when the sample size is sufficiently large, the distribution of sample mean is approximated to normal distribution. How large will be sufficiently enough? Some researchers may use the criterion: in practical applications, the distribution of sample mean may be assumed to be normal distribution if the sampling size is larger than 30. But various shapes of probability distributions exist, e.g. single peak and multi-peak, symmetric and asymmetric, high skewness and low skewness, and also, the uniform distribution with symmetry but no peak, no skewness and no tails. Furthermore, there are distributions similar to the normal distribution while the others are vastly different. The purpose of this thesis is to examine the criterion mentioned above by simulation. We consider four continuous distributions in chapter 2 through chapter 5, including uniform, triangular, gamma and Weibull distributions and have provided regression models and tables of the required sample size in using central limit theorem.
In investigation interview, interviewees feel panic of privacy to be disclosed on sensitive subjects that they often refuse or untrue to answer the sensitive questions. In chapter 6, some indirect randomized response techniques are proposed, which maintain the requirement of efficiency and protection of confidentiality. The interviewee is only required to report a positive or negative integer, something that every individual participating in a survey is expected to be capable of. Since the information provided to the interviewer is not sufficient to verify whether an individual possesses the characteristic or not, the respondents’ privacy is well protected. In this regard, the respondents are perhaps more willing to cooperate and report truthfully. For the sake of simplicity of survey process, the proposed procedure seems more practicable than Christofides (2003) procedure. In section 6.2, we also consider the decision of sample size in using the above indirect randomized response techniques. Three sets of finite discrete distributions are utilized to demonstrate the application of central limit theorem.
論文目次 表目錄......................................................................................................三
圖目錄......................................................................................................五
使用符號一覽表......................................................................................七
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 本文結構...................................................................................7
第二章 中央極限定理於均勻分配上之樣本數探討............................9
  2.1 標準形態均勻分配之模擬.......................................................9
  2.2 一般區間均勻分配之應用.....................................................13
第三章 中央極限定理於三角分配上之樣本數探討..........................15
  3.1 標準形態三角分配之模擬.....................................................15
  3.2 一般化三角分配之應用.........................................................24
第四章 中央極限定理於伽瑪分配上之樣本數探討..........................26
  4.1 模擬.........................................................................................26
  4.2 分析.........................................................................................40
第五章 中央極限定理於韋伯分配上之樣本數探討..........................49
  5.1 模擬.........................................................................................49
  5.2 分析.........................................................................................61
第六章 中央極限定理於隨機回答調查上之樣本數探討..................73
  6.1 隨機回答調查之設計.............................................................73
  6.2 中央極限定理之應用.............................................................76
第七章 結論..........................................................................................84
  7.1 主要研究結果.........................................................................84
  7.2 未來研究方向.........................................................................88
參考文獻..................................................................................................89

表 目 錄
表2.1 均勻分配之W檢定結果............................................................10
表2.2 均勻分配之倒數迴歸模式.........................................................11
表3.1 三角分配之W檢定結果............................................................16
表3.2 三角分配之倒數迴歸模式.........................................................22
表3.3 三角分配符合中央極限定理所需之最小樣本數.....................23
表4.1 伽瑪分配之W檢定結果............................................................28
表4.2 伽瑪分配之倒數迴歸模式.........................................................43
表4.3 伽瑪分配在不同要求拒絕率下之倒數迴歸模式.....................47
表4.4 伽瑪分配應用中央極限定理所需之最小樣本數.....................47
表5.1 韋伯分配之W檢定結果一........................................................51
表5.2 韋伯分配之W檢定結果二........................................................53
表5.3 韋伯分配之曲線迴歸模式.........................................................63
表5.4 韋伯分配之倒數迴歸模式一 ( 6 . 3 ≤ c )....................................67
表5.5 韋伯分配之倒數迴歸模式二 ( 6 . 3 ≥ c )....................................67
表5.6 韋伯分配應用中央極限定理所需之最小樣本數.....................71
表6.1 隨機裝置之對應數值.................................................................74
表6.2 三組有限離散型分配之W檢定結果........................................78
表6.3 三組有限離散型分配之指數迴歸模式.....................................82
表6.4 三組有限離散型分配應用中央極限定理所需之最小樣本數
....................................................................................................83


圖 目 錄
圖1.1 研究結構.......................................................................................8
圖2.1 在均勻分配下,樣本數與拒絕常態性檢定次數m 之關係.......11
圖2.2 均勻分配關係示意.....................................................................13
圖3.1 在三角分配下,樣本數與拒絕常態性檢定次數m 之關係.......21
圖3.2 三角分配關係示意.....................................................................24
圖4.1 在伽瑪分配下,樣本數與拒絕常態性檢定次數m 之關係.......41
圖4.2 伽瑪分配之倒數迴歸模式曲線.................................................42
圖4.3 伽瑪分配在不同要求拒絕率下之伯努利試驗結果示意.........45
圖4.4 伽瑪分配之參數α與樣本數之倒數迴歸模式曲線.................46
圖5.1 在韋伯分配下,樣本數與拒絕常態性檢定次數m 之關係.......62
圖5.2 韋伯分配在不同要求拒絕率下之伯努利試驗結果示意(一)
....................................................................................................64
圖5.3 韋伯分配在不同要求拒絕率下之伯努利試驗結果示意(二)
....................................................................................................65
圖5.4 韋伯分配在要求拒絕率5 = ′ m 下,參數c 與樣本數之倒數迴歸
模式曲線....................................................................................68
圖5.5 韋伯分配在要求拒絕率10 = ′ m 下,參數c 與樣本數之倒數迴歸

模式曲線....................................................................................69
圖5.6 韋伯分配在要求拒絕率20 = ′ m 下,參數c 與樣本數n 之倒數迴
歸模式曲線................................................................................70
圖6.1 在有限離散型分配下,樣本數與拒絕常態性檢定之次數m 之
關係............................................................................................80
圖6.2 在有限離散型分配下,樣本數與拒絕常態性檢定之次數m 之
指數迴歸曲線............................................................................81
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