The Effect of Truncation on Tests of Hypotheses for Normal Populations

Abstract

When the means of samples drawn from normal populations are employed in performing tests of hypothesis, the experimenter usually assumes that the sample was drawn from a complete normal population. When all members of the population within three standard deviations of the mean admit to sampling, assumption that the sample was drawn from a complete population does not seriously affect the power of the tests. However, there arise situations where the sampling interval is restricted to less than three standard deviations of the mean. Aggarwal and Guttman [1], [2] have examined the loss of power when using tests based on the assumption that the variable being sampled has a complete normal distribution when, in fact, the distribution is a symmetrically truncated normal distribution. They derived the distribution of means based on samples of size $n \leqq 4$ using convolutions of \begin{align*}\tag{1.1}f(x) &= (C/(2\pi)^{\frac{1}{2}}) \exp(-x^2/2), |x| < a, \\ &= 0,\quad\text{otherwise},\end{align*} where $C$ is given by \begin{equation*}\tag{1.2}C^{-1} = (2\pi)^{-\frac{1}{2}} \int^a_{-a} e^{-\frac{1}{2}t^2} dt.\end{equation*} Birnbaum and Andrews [2] have pointed out that sums of symmetrically truncated normal random variables have a limiting normal distribution. Thus for large $n$ one can obtain for an approximate cumulative distribution of $\bar X = (1/n) \sum^n_{i = 1} X_i$. For arbitrary $n$, however, no general formula giving the distribution of means (or sums) of samples of size $n$ drawn from a truncated normal population is available. In this paper we extend the work of Aggarwal and Guttman [1] to non-symmetric truncation and arbitrary sample size. An asymptotic series for the distribution of sums of samples of size $n$ drawn from a truncated normal population is presented.