淡江大學機構典藏:Item 987654321/33207
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    Title: 中央極限定理的樣本數模擬探討
    Other Titles: On sample size simulation in central limit theorem
    Authors: 吳昭賢;Wu, Chao-hsien
    Contributors: 淡江大學管理科學研究所博士班
    張紘炬;Chang, Horng-jinh
    Keywords: 樣本數;亂數;模擬;Shapiro-Wilk;常態性檢定;隨機回答;Sample Mean;Random Number;simulation;Shapiro-Wilk;Normality Test;Randomized Response
    Date: 2005
    Issue Date: 2010-01-11 03:44:10 (UTC+8)
    Abstract: 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。
    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.
    Appears in Collections:[Department of Management Sciences] Thesis

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