Abstract

This study was undertaken to lay down some definite criterion for the evaluation of the precision of abnormal range determined by means of a random sample form a normal population. There are two types of abnormal range. In the first type an abnormal range is set up unilaterally and in the second type bilaterally in a normal distribution. In the first type an abnormal range sized 100α% is assumed to be set up on the right side in a normal distribution N(μ, δ²). In the parametric method to determine the normal range is to estimate the lower limit of the abnormal range sized 100α% by

were and s are the mean and standard deviation of a n-sized random sample from a normal population N(μ, δ²) and K_α is the value defined as follows:
where Φ(u) is the standardized normal distribution N(0.1²). Y₁is a random variable which varies from sample to sample and is distributed normally with mean
and variance
since and s are stochastically independent and the former is distributed normally with parameters (μ, δ²/n) and the latter with parameters (c₁δ, (c₂δ)²), where c₁=1-1/4n and c₂=1/. Then, the upper and lower limits of the middle 100p%-range of Y₁denoted by Y₁₍₊₎(Χ_p) and Y_1(-)(Χ_p) are easily obtained by the following formulas:
where Χ_p is the square root of 100p%-fractile of Χ²-distribution with d.f.1, that is, Χ_p²(1) and . Now, let us estimate the 100α%-size of the abnormal range by u₁₍₊₎(Χ_p) and u_1(-)(Χ_p) which are obtained by the following transformation,
The estimates of the 100α%-size of the abnormal range by Y₁₍₊₎(Χ_p) and Y_1(-)(Χ_p) denoted by â₁(Χ_p) and â_1(-)(Χ_p) correspondingly are
Hence
where
Accordingly, it can be stated that 100p% of the estimated values of the 100α%-size of the right abnormal range are between â₁₍₊₎(Χ_p) and â_1(-)(Χ_p). By the same argument it could easily be varified that 100p% of the estimated values of the same size of a left abnormal range are between the same limits. It is evident that by the formula(1.7.1) we can evaluate the precision of the estimate by a n-sized random sample of the 100α%-size of the abnormal range set up unilaterally in a normal distribution. Now the second type of normal range is considered. This type is assumed to be characterized by bilateral abnormal ranges with equal size of 100α% set up on both the right and the left side in a normal distribution N(μ, δ²). In this case the upper and lower limits of the normal range are estimated
where and s are the mean and standard deviation of a n-sized random sample which are distributed normally with parameters (μ, δ²/n) and (c₁δ, (c₂δ)²), correspondingly. So the point Y(,s) follws a two dimensional normal distribution with correlation coefficient o. If the closed domain which is defined by the contour ellipse, E(Χ_p), corresponding to the probability p determined by Χ_p²(2) in the s-plane is denoted by
A(Χ_p),
where Χ_p is the square root of Χ_p²(2). Now let us standardize the two variable and s by the follwing transformation,
then the point U(u₁, u₂) follows the standardized two dimensional normal distribution with correlation coefficient o. If the closed domain which is defined by the contour ellipse, E’(Χ_p)_γ corresponding to the probability p determined by Χ_p²(2) in the u₁u₂-plane is denoted by A’(Χ_p)_γ again
Here it is worthy of note that there is one to one correspondance between Y∈A(Χ_p) and U∈A’(Χ_p) and that E’(Χ_p) is a circle with radius Χ_p and center (0,0) in the u₁u₂-plane. By the inverse transformation of (2.3)
Substituting these for and s in the equiations (2.1.1) and (2.1.2)
For the estimation of the 100α%-size of each abnormal range set up bilaterally, it is necessary to standardize
Y_(±) by using μ and δ as follows:
where the symbols t_(±)(u₁, u₂) indicate the standardized deviates of Y_(±). If the estimates of the 100α%-size of the abnormal range set up bilaterally by Y∈A(Χ_p) and U∈A’(Χ_p) that corresponds to Y∈A(Χ_p) are denoted by â_(±)(Y∈A(Χ_p) and â_(±)(U∈A’(Χ_p), correspondingly, then
By summing (2.8.1) and (2.8.2) a new function of U(u₁, u₂),
is introduced to estimate the 100(2α)%-size of the abnormal ranges. Since E’(Χ_p) is a circle with radius Χ_p and ceter (0,0) t_(±)(U∈E’(Χ_p))-equivalent to t_(±)((u₁, u₂)∈E’(Χ_p))-can be expressed in terms of polar coordinates by putting u₁=Χ_pcosΘ and u₂=Χ_psinΘ as in the following:
By substituting these for the end values t_(±)(u₁, u₂) in the integrals (2.8.1), (2.8.2) and (2.8.3), these become
and
Examining the variation of the function of Θ, â(U∈E’(Χ_p), it is found that the function has the minimum value at the point (Χ_p, π/2)∈E’(Χ_p) and the maximum value at the point (Χ_p, 3π/2)∈E’(Χ_p) and that
that is,
Now it is evident that by the formula (2.11.1) we are able to evaluate the precision of the estimate by a n-sized random sample of 100(2α)%-size of the abnormal ranges set up bilaterally in a normal distribution. At the end of this paper, tables are provided which are available for the evaluation of the precision of estimates of normal or abnornal ranges by random samples from normal populations.