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.