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# On Normal Approximation of χ2 Distribution

Horng-Jinh Chang This email address is being protected from spambots. You need JavaScript enabled to view it.1 and Ming-Chen Lee1

1Department of Management Sciences, Tamkang University, Tamsui, Taiwan 25137, R.O.C.

Accepted: July 25, 2017
Publication Date: September 1, 2017

## ABSTRACT

According to the central limit theorem, if χ1, χ2, …χυ is a random sample drawn from χ2(1), then, when , the distribution function of the sample mean χ=∑χi/υ would asymptotically approximate to N (1,2/υ), or the distribution function of (X -1)/√(2/υ) would approximate to the standard normal distribution N(0, 1). Also, the distribution function of X i i 1 would asymptotically approximate to the normal distribution N(υ, 2υ). Many statistics textbooks or applied statistics research accept the use of a sample size of  υ≥30 for the assumption of N(υ, 2υ) approximating to X . Therefore, in the present study, computer simulation was adopted to test the required sample size for the normal distribution to approximate to the χ2 distribution. This information is useful for the applications of the central limit theorem.

Keywords: Computer Simulation, The Central Limit Theorem, χ2 Distribution, Normal Distribution

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