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International Journal of Statistical Sciences ISSN 1683–5603
Vol. 25(1), March, 2025, pp 11-22
DOI: https://doi.org/10.3329/ijss.v25i1.81041
©
2025 Dept. of Statistics, Univ. of Rajshahi, Bangladesh
Optimal Prediction and Tolerance Intervals for the Ratio
of Dependent Normal Random Variables
K. Krishnamoorthy1∗and Gboyega Adepoju2
1,2Department of Mathematics, University of Louisiana at Lafayette, USA
∗Correspondence should be addressed to K. Krishnamoorthy
(Email: krishna@louisiana.edu)
[Received December 25, 2024; Accepted February 20, 2025]
Abstract
A simple exact method is proposed for computing prediction intervals and
tolerance intervals for the distribution of the ratio X1/X2when (X1, X2)
follows a bivariate normal distribution. The methodology uses the factors
available for computing one-sample prediction intervals and tolerance inter-
vals for a univariate normal distribution. Both one-sided and two-sided inter-
vals are constructed, and the two-sided tolerance intervals are obtained with
and without imposing the equal-tail requirement. The results are illustrated
using two practical applications that call for the computation of prediction
intervals and tolerance intervals for the distribution of the ratio X1/X2. The
first application is an investigation of retroviral contamination in the raw
materials used for the manufacture of the influenza vaccine FluMist. The
second application is on the cost-effectiveness of a new drug compared to a
standard drug.
Keywords: Equal-tailed tolerance interval, Non-central t, Prediction factor,
Tolerance factor.
AMS Classification: 62F25.
1. Introduction
This work is motivated by two applications, available in the literature, that call for
the computation of prediction intervals and tolerance intervals for the ratio of the
12 International Journal of Statistical Sciences, Vol. 25(1), 2025
marginal random variables in a bivariate normal distribution. The first application is
an investigation of retroviral contamination in the raw materials used the manufacture
of the influenza vaccine FluMist, and the second application is on the cost-effectiveness
of a new drug compared to a standard drug.
A major component of statistical data analysis is the computation of appropriate
intervals, motivated by specific applications. These intervals include confidence inter-
vals, prediction intervals and tolerance intervals. As is well known, such intervals can
be computed under both parametric and non-parametric scenarios. We recall that
a prediction interval is an interval computed using a random sample, satisfying the
condition that the interval will include the future values of a random variable with a
given confidence level (say, γ). On the other hand, a tolerance interval for a distri-
bution is an interval computed using a random sample, subject to the condition that
the interval includes a specified proportion (say, p) or more of the distribution, with a
specified confidence level (say, γ). The proportion passociated with a tolerance inter-
val is referred to as its content and the interval is often referred to as a (p, γ) tolerance
interval. Prediction intervals and tolerance intervals could be one-sided, with only
a lower limit or only an upper limit, or they could be two-sided. A (p, γ) tolerance
interval has several practical applications; the book by Krishnamoorthy and Mathew
(2009) provides a detailed treatment of the topic.
The present work is on the computation of prediction intervals and tolerance intervals
in a specific parametric scenario, namely, a bivariate normal setup where interest is
focused on the ratio of the random variables. Let X= (X1, X2)′be a bivariate normal
random vector with mean µand variance-covariance matrix Σ. We want to compute
one-sided and two-sided prediction intervals and tolerance intervals for X1/X2based
on a sample. If we write µ= (µ1, µ2)′, the computation of a confidence interval for
µ1/µ2is a well-investigated problem. However, we are not aware of any work that
addresses the computation of prediction intervals for X1/X2. Furthermore, only very
limited literature is available for computing tolerance intervals for X1/X2; see Zhang
et al. (2010) and Flouri et al. (2017). An earlier article by Hall and Sampson (1973)
provides an approximate solution assuming that the coefficients of variation are small;
the paper also includes an application related to drug development. In their works,
Zhang et al. (2010) and Flouri et al. (2017) discuss applications relevant to bioassays
and cost-effectiveness analysis where it is necessary to compute tolerance intervals for
a ratio random variable of the type X1/X2. Ratio random variables also arise in the
context of process control problems in manufacturing applications; see Celano et al.
(2014), Celano and Castagliola (2016) and Tran et al. (2021) for details and other
relevant references. Furthermore, tolerance limits have been recommended to be used
as process control limits; see Hamada (2003) and Alqurashi (2021).
The methodology in Zhang et al. (2010) is based on the generalized variables approach,
and the authors address the development of a one-sided tolerance limit only. Here we
shall not elaborate on their approach. On the other hand, Flouri et al. (2017) address
the computation of both one-sided tolerance limits and two-sided tolerance intervals;
Krishnamoorthy and Adepoju: Optimal Prediction and Tolerance Intervals. . . 13
they appeal to non-parametric methods for computing the tolerance limits based on
a parametric bootstrap sample, and then apply a bootstrap calibration so as to have
better coverage probabilities. Thus the methodology is computationally demanding.
Our work presents a simple and accurate approach for computing exact one-sided
and two-sided prediction intervals and tolerance intervals for the ratio X1/X2in the
bivariate normal scenario. Our intervals are in fact exact; i.e., they guarantee the
coverage probability. However, our approach produces bona fide prediction limits (or
tolerance limits) only when the marginal lower prediction limits (respectively, marginal
lower tolerance limits) are positive. We expect this to be the case when the support of
the marginal distributions is mostly on the positive part of the real line. In addition
to deriving the prediction intervals and tolerance intervals, we have also illustrated
our methodology by applying it to the two applications that motivated our work, and
mentioned at the beginning of this section.
2. Prediction Intervals and Tolerance Intervals
Let X1, ..., Xnbe a sample from a bivariate normal distribution with mean vector µ
and variance-covariance matrix Σ,N2(µ,Σ). Furthermore, let ¯
Xand Sdenote the
sample mean vector and sample covariance matrix, respectively. Write
µ=µ1
µ2,Σ=σ2
1ρσ1σ2
ρσ1σ2σ2
2,¯
X=¯
X1
¯
X2and S=S2
1bρS1S2
bρS1S2S2
2,
where ρand bρ, respectively, denote the population correlation coefficient and the
sample correlation coefficient. We start with a derivation of our proposed prediction
interval for X1/X2when X= (X1, X2)′∼N2(µ,Σ).
2.1. Prediction Intervals
A 100γ% prediction interval (PI) for X1/X2, say L(k;¯
X,S), U(k;¯
X,S)satisfies
P¯
X,S,X1,X2L(k;¯
X,S)≤X1
X2≤U(k;¯
X,S)=γ, (1)
where kis the prediction factor to be determined subject to the above condition. We
note that the condition (1) is based on the joint distribution of ( ¯
X,S, X1, X2).It
turns out that a PI for X1/X2can be deduced from the one for X1−RX2, where
Ris a constant (to be determined). Let ¯
Yand S2
ydenote the sample mean and
sample variance, respectively, based on Yi=X1i−RX2i,i= 1, ..., n, where we write
Xi= (X1i, X2i)′. A 100γ% PI for a future observation Y∗=X∗
1−RX∗
2is given by
¯
Y±kSy=¯
X1−R¯
X2±kqS2
1−2RbρS1S2+R2S2
2,(2)
14 International Journal of Statistical Sciences, Vol. 25(1), 2025
where k=tn−1;(1+γ)/21 + 1
n1/2, and tm;αdenotes the 100αpercentile of the tdistri-
bution with degrees of freedom (df) m. It should be clear that the quantity under the
square root sign in (2) is simply the estimated variance of X1−RX2. Equating the
two endpoints of the above interval to zero, and solving the resulting equations for R,
we find prediction limits for X1/X2. The limits are given by
(L(k;¯
X1,¯
X2, S1, S2), U (k;¯
X1,¯
X2, S1, S2))
=
¯
X1¯
X2−k2bρS1S2∓q¯
X1¯
X2−k2bρS1S22−¯
X2
1−k2S2
1¯
X2
2−k2S2
2
¯
X2
2−k2S2
2.(3)
Clearly, the above limits provide a bona fide prediction interval only when the quantity
under the square root sign is positive. As shown in Appendix A, a sufficient condition
for this is that the marginal lower prediction limits ¯
X1−kS1and ¯
X2−kS2are both
positive. We have also shown in Appendix B that the lower prediction limit is more
likely positive when both X1and X2are positive random variables. Furthermore, the
PI is equivariant under the scale transformations X1→c1X1and X2→c2X2and it
is invariant when c1=c2=c. We also note that the upper prediction limit for X2/X1
is obtained as the reciprocal of the lower prediction limit of X1/X2. That is,
U(k;¯
X2,¯
X1, S2, S1) = 1/L(k;¯
X1,¯
X2, S1, S2),(4)
where U(k;¯
X2,¯
X1, S2, S1) is a 100γ% upper prediction limit for X2/X1. Noticing that
U(k;¯
X2,¯
X1, S2, S1) and U(k;¯
X1,¯
X2, S1, S2) differ only in the denominator terms, mul-
tiplying both sides by L(k;¯
X1,¯
X2, S1, S2) and using the result that (a−b)(a+b) =
a2−b2the above relation can be readily verified.
2.2. One-Sided Tolerance Limits
A one-sided (p, γ) upper tolerance limit (TL) for the distribution of the ratio X1/X2
is a quantity U(¯
X,S) satisfying
P¯
X,SPX1
X2≤U(¯
X,S)(¯
X,S)≥p=γ.
It is known that a (p, γ) upper TL for any distribution is simply a 100γ% upper
confidence limit for the pth quantile of the distribution. In view of this, if Rp(µ,Σ) is
the pth quantile of the distribution of X1/X2, then U(¯
X,S) also satisfies
P¯
X,SU(¯
X,S)≥Rp(µ,Σ)=γ.
Notice that for a constant R,X1−RX2∼N(µ1−Rµ2, σ2
R), where σ2
R=σ2
1−
2Rρσ1σ2+R2σ2
2, and so its 100ppercentile is given by µ1−Rµ2+zpσR, where zp
is the pth quantile of the standard normal distribution. An upper confidence limit
Krishnamoorthy and Adepoju: Optimal Prediction and Tolerance Intervals. . . 15
for Rp(µ,Σ), the pth the percentile of X1/X2, can be deduced from an upper CL for
µ1−Rµ2+zpσR. Specifically, by equating the upper CL for µ1−Rµ2+zpσRto zero
and then solving the resulting equation for R, we can get an upper confidence limit
for Rp(µ,Σ). Let
k1=1
√ntn−1;γ(zp√n),(5)
where tm;α(δ) denotes the αquantile of the noncentral tdistribution with df = mand
the noncentrality parameter δ. It is known that an upper confidence limit for the pth
percentile of a normal distribution is given by
(sample mean) + k1×SD.
For example, see Krishnamoorthy and Mathew (2009, Chapter 2). Thus, a 100γ%
upper confidence limit for µ1−Rµ2+zpσDis given by
¯
X1−R¯
X2+k1qS2
1−2RbρS1S2+R2S2
2.(6)
Note that the above upper confidence limit is similar to the prediction limit in (2)
except for the factor. Therefore, equating (6) to zero, and solving the resulting equa-
tion for R, we will get a 100γ% upper confidence limit for Rp(µ,Σ), which is the
pth quantile of the distribution of X1/X2. In other words, such a solution for R
is a (p, γ) upper TL for the distribution of X1/X2. The tolerance limit is given by
U(k1;¯
X1,¯
X2, S1, S2), similar to (3), except that a different factor is used. Similarly, a
(p, γ) LTL can be obtained as L(k1;¯
X1,¯
X2, S1, S2).These one-sided TLs also possess
some natural properties like the prediction limits. They are bona fide limits provided
individual lower TLs ¯
X1−k1S1and ¯
X2−k1S2are positive and equivariant under
scale transformations; see appendices A and B. Furthermore, an upper TL for the
distribution of X2/X1can be obtained as reciprocal of the lower TL for X1/X2. In
other words,
U(k1;¯
X2,¯
X1, S2, S1) = 1/L(k1;¯
X1,¯
X2, S1, S2).
2.3. Two-Sided Tolerance Intervals
There are two types of two-sided tolerance intervals. A two-sided tolerance interval
denoted by L(kt;¯
X1,¯
X2, S1, S2), U (kt;¯
X1,¯
X2, S1, S2)is constructed so that it would
include at least a proportion pof the population with confidence γ. That is,
P¯
X,SPX1,X2L(kt;¯
X,S)≤X1
X2≤U(kt;¯
X,S)¯
X,S≥p=γ, (7)
where ktis the factor to be determined so as to satisfy the probability requirement
in (7). The second type of two-sided tolerance intervals are called equal-tailed toler-
ance intervals or central tolerance intervals. Such intervals are constructed subject to
16 International Journal of Statistical Sciences, Vol. 25(1), 2025
the condition that the interval would include the lower 100(1 −p)/2 percentile and
the upper 100(1 −p)/2 percentile with probability γ. In other words, the interval
would include the central 100p% of the distribution. Thus an equal-tailed interval, say
L(ke;¯
X1,¯
X2, S1, S2), U (ke;¯
X1,¯
X2, S1, S2), satisfies
P¯
X,SnL(ke;¯
X1,¯
X2, S1, S2)≤R1−p
2(µ,Σ) and R1+p
2(µ,Σ)≤U(ke;¯
X1,¯
X2, S1, S2)o=γ,
(8)
where Rα(µ,Σ) denotes the 100αpercentile of the distribution of X1/X2, and keis
the factor to be determined subject to the above probability requirement.
Since Y=X1−RX2follows a univariate normal distribution when Ris a constant, the
two-sided tolerance factors ktand kecan be obtained from what is already available
for the normal distribution; see Krishnamoorthy and Mathew (2009, Chapter 2). For
instance, let ¯
Y±kpcvar(Y) be a two-sided TI for the distribution of Y, where k=kt
for the TI defined in (7) and k=kefor the TI defined in (8). Since Yi’s are independent
normal random variables, ktis the factor for computing a TI for a normal distribution,
and is the solution of the integral equation
r2n
πZ∞
0
P χ2
m>mχ2
1;p(z2)
k2
t!e−1
2nz2dz =γ, (9)
where m=n−1; see Krishnamoorthy and Mathew (2009, Section 2.3). The two-sided
TI is given in (3) with kreplaced by kt.
Similarly, the factor kefor constructing equal-tailed TI of the form ¯
Y±kepcvar(Y) is
the solution of the integral equation
1
2m
2Γ(m
2)Z∞
mδ2
k2
en2Φ −δ+ke√nx
√m−1e−x/2xm
2−1dx =γ, (10)
where Φ(x) denotes the standard normal distribution function. Details are once again
given in Krishnamoorthy and Mathew (2009, Section 2.3). The equal-tailed TI is given
in (3) with kreplaced by ke.
A sufficient condition for the existence of these two-sided TIs is that
¯
Xi−factor ×Si>0,for i= 1,2,
where the factor is ktor ke. In Appendix B, we provided some numerical evidence
that the above conditions hold with high probability. The Rpackage by Young (2020)
or the PC calculator StatCalc that accompanies the book by Krishnamoorthy (2016)
can be used to compute the factors ktand ke.
3. Examples
We shall now present two examples in order to illustrate our methodology for comput-
ing prediction intervals and tolerance intervals for the ratio X1/X2. The first example
Krishnamoorthy and Adepoju: Optimal Prediction and Tolerance Intervals. . . 17
is discussed in Zhang et al. (2010), and both the examples are given in Flouri et al.
(2017).
Example 1. This application described in Zhang et al. (2010) is in the context of
the manufacture of the influenza vaccine FluMist, and is an investigation of retroviral
contamination in the raw materials used in its manufacture. The reverse transcriptase
(RT) assay is used to detect contamination, and the presence of the enzyme RNA
directed DNA polymerase is indicative of the presence of the retrovirus. Additionally,
a large radioactivity count in the RT assay is taken as evidence for a large amount of
the enzyme in the sample. A sample is classified as negative by comparing the radioac-
tivity count from the sample with that from a negative control. Operationally, this
amounts to verifying if the ratio of the two radioactivity counts is below a threshold.
Thus the two radioactivity counts is the random variable of interest, and historical
in-control data supports the bivariate normality assumption. For more details, we
refer to the article by Zhang et al. (2010).
The bivariate sample under consideration consists of n= 45 pairs of radioactivity
counts in the negative controls and in the in-control RT assays. The Shapiro-Wilk
normality test by Zhang et al. (2010) revealed that the data were distributed as
bivariate normal. Furthermore, the sample data gave the following statistics (sample
means ¯x1and ¯x2, sample variances s2
1and s2
2, and sample correlation coefficient ˆρ):
¯x1= 38.1, s2
1= 56.3,¯x2= 38.9, s2
2= 35.1,bρ= 0.81.
Let X1and X2denote the radioactivity counts in the negative controls and in the in-
control RT assays, respectively. The factor for computing a 95% two-sided prediction
interval for the ratio X1/X2is obtained as tn−1;1−α/2p1+1/n =t44;0.975p1+1/45 =
2.038 and the 95% PI (3) is (0.731, 1.218). Similarly, we computed the 99% PI as
(0.631, 1.309).
In order to construct a (.90, .95) upper tolerance limit for X1/X2, we first find the
factor as t44;.95(z.9√45)/√45 = 2.0924. We computed the upper tolerance limits and
have reported them in Table 1 along with those based on the two generalized vari-
able methods by Zhang et al. (2010); in the table, these are denoted by GVI and
GVII. Compared to the factors based on the generalized variable approach, the exact
approach has produced factors that are somewhat smaller, and hence these are to be
preferred. However the difference between the tolerance factors based on the two ap-
proaches is somewhat insignificant. We recall that the Zhang et al. (2010) approach
can produce only one-sided tolerance limits.
The factors for computing (p, .95) two-sided tolerance interval are 2.4116 and 3.1680
when p= 0.95 and 0.99, respectively. The factors for computing a (p, .95) equal-
tailed tolerance interval are 2.5595 and 3.3005 when p= 0.95 and 0.99, respectively.
18 International Journal of Statistical Sciences, Vol. 25(1), 2025
The corresponding two-sided intervals are also given in Table 1. As expected, the
equal-tailed tolerance interval is wider compared to the tolerance interval that does
not impose this requirement.
Table 1: (p, .95) upper TLs and tolerance intervals for the distribution of X1/X2for
Example 1
Exact method
pGV I GV II One-sided Two-sided Equal-tailed
0.95 1.233 1.230 1.224 (0.678, 1.266) (0.656, 1.286)
0.99 1.346 1.343 1.334 (0.555, 1.376) (0.531, 1.397)
Example 2. Ratio parameters and ratios of random variables are encountered in ap-
plications dealing with cost-effectiveness analysis. Our focus here is on a specific
application in cost-effectiveness analysis where the problem of interest is the computa-
tion of tolerance intervals for a ratio random variable applying our methodology. Thus
we shall not give a discussion of the area of cost-effectiveness analysis and the relevant
statistical methodologies; for details we refer to the book by Willan and Briggs (2006).
Our example involves the comparison of two pharmacological agents (a test drug and a
reference drug), and is adapted from Gardiner et al. (2000). In this comparison trial,
each of the test drug and the reference drug was administered to 150 patients, and
the effectiveness of the drug was assessed using Quality-Adjusted-Life-Years (QALYs).
Furthermore, the cost of the treatment was in US dollars. This application first ap-
peared in Sacristan et al. (1995), and the data was later analyzed in Gardiner et al.
(2000) and Flouri et al. (2017). Bivariate normality was assumed by these authors
for the cost (C) and effectiveness (E) and the problem of interest is the computation
of tolerance limits for C/E for the test drug and for the reference drug. As noted by
Flouri et al. (2017), the original article by Sacristan et al. (1995) does not report the
correlation between the cost and effectiveness. In the absence of correlations, Gardiner
et al. (2000) assumed the value 0.7 for the correlation coefficients bρiin both groups.
In their work, Flouri et al. (2017) used the same value for the correlation coefficients
in both groups and illustrated their methods of constructing TIs for C/E. The means
and standard deviations for the two groups are as follows:
Table 2: Summary statistics for Example 2
Sample Mean Mean SD SD
size cost effectiveness cost effectiveness bρ
Test drug 150 200000 8 78400 2.1 0.7
Reference drug 150 80000 5 27343 2
The factor for computing 90% one-sided PIs for the ratio X1/X2is 1.292. For the test
Krishnamoorthy and Adepoju: Optimal Prediction and Tolerance Intervals. . . 19
drug, the 90% one-sided lower and upper prediction limits are 15231.0 and 34474.4,
respectively. For the reference drug, the 90% one-sided lower and upper prediction
limits are 10907.2 and 25779.1, respectively. These one-sided prediction limits are
computed using (3).
The one-sided and two-sided tolerance limits for the same example are given in Table
3 and Table 4, respectively. Between the test drug and the reference drug, we notice
significant differences among the one-sided tolerance limits, and also among the two-
sided tolerance intervals. In fact we note larger tolerance limits for the distribution of
C/E for the test drug, compared to those for the standard drug in the case of both
one-sided tolerance limits as well as two-sided tolerance intervals. These differences
indicate that the two drugs are not similar. This is also the conclusion in Flouri et al.
(2017).
Table 3: (0.90, 0.95) one-sided tolerance limits for the ratio C/E for the test drug and
for the reference drug for Example 2
Method Bootstrap Tolerance Exact method
LTL UTL factor LTL UTL
Test drug 14262.37 35639.61 1.478 13566.16 36032.33
Reference drug 10058.54 27822.01 1.478 10216.08 28690.79
Table 4: (.90,.95) tolerance intervals for the ratio C/E for the test drug and for the
reference drug for Example 2
Exact method
Method Bootstrap TI Two-sided Equal-tailed
Test drug (9441.55, 40017.71) (10084.12, 39239.81) (9114.08, 40121.31)
Reference drug (8505.69, 39680.02) (8849.58, 37854.98) (8481.67, 41698.69)
4. Discussion
Interval estimation problems involving ratio of parameters and ratio of random vari-
ables present challenges when it comes to developing the necessary methodology that
guarantees the desired statistical properties; for example, satisfying the coverage prob-
ability requirements and having solutions that are indeed finite intervals. While the
interval estimation of the ratio of normal means is a well investigated problem, only
very limited literature is available for computing prediction intervals and tolerance in-
tervals for the ratio random variable in a bivariate normal distribution. This article is
on the development of such intervals. Some attractive features of our intervals are that
they are exact, and are very easy to compute since they have explicit analytic expres-
sions. It is hoped that they will be used in applications that call for the computation
of prediction limits and tolerance limits for ratio random variables; for example, in
statistical process control problems that are ratio based. However, we have not taken
up such applications in the present work.
20 International Journal of Statistical Sciences, Vol. 25(1), 2025
Acknowledgements
The authors are grateful to two reviewers for providing useful comments and sugges-
tions.
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Appendix A
To show that the term T=¯
X1¯
X2−k2bρS1S22−¯
X2
1−k2S2
1¯
X2
2−k2S2
2under
the radical sign in (3) is positive, let ai=¯
Xi/Si,i= 1,2. It is easy to see that the
above term is positive if
(a1a2−k2bρ)2−(a2
1−k2)(a2
2−k2)>0.
After simplification, we see that the above condition simplifies to (a2
1+a2
2) + k2bρ2−
k2−2bρa1a2>0. Using the result a2
1+a2
2>2a1a2, it is sufficient to show that
2a1a2(1 −bρ)−k2(1 + bρ)(1 −bρ)>0.
Noting that bρ < 1, a sufficient condition for the above inequality to hold is a1> k
and a2> k. That is, ¯
X1−k√S11 >0 and ¯
X2−k√S22 >0.Also, under these
conditions, the denominator term ¯
X2
2−k2S2
2in (3) is also positive because ¯
X2
2−k2S2
2=
(¯
X2−kS2)( ¯
X2+kS2).
Appendix B
We now show that the ¯
X1−kS1is positive with high probability. We first note that
¯
X1−kS1>0 if and only if
¯
X1
S1
> k = (1 + 1/n)1/2tn−1;1+γ
2
.
It is easy to show that ¯
X1
S1∼1
√ntn−1n1
µ1
σ1.
22 International Journal of Statistical Sciences, Vol. 25(1), 2025
Since we postulate normal model for a positive random variable X1, the population
parameters are expected to satisfy µ1−3.3σ1>0, or equivalently, µ1/σ1>3.3. Then
P(¯
X1/S1> k) = Ptn−1(√nµ1/σ1)>√n+ 1tn−1; 1+γ
2
> P tn−1(3.3×√n)>√n+ 1tn−1; 1+γ
2
=P(n, γ),say.
To get the above inequality, we used that fact that the noncentral tdistribution is
stochastically increasing in the noncentrality parameter. The above probability is very
close to 1 for many practical values of nand γ. For example, P(n, .95) = 0.987, .997
and 0.9999 when n= 15,20 and 30, respectively.
To show that the one-sided tolerance limits are also more likely positive, we need to
assess P(¯
X1/S1> k1), where k1=1
√ntn−1;γ(zp√n); see Eqn (5). As argued earlier for
the prediction limits, this probability
P(¯
X1/S1> k1) = P¯
X1/S1>1
√ntn−1;γ(zp√n)
> P tn−1(3.3×√n)> tn−1;γ(zp√n)
=P1(n, p, γ),say.
The above probability is also close to 1 for many reasonable cases. For example,
P1(n, .90, .95) = .996 and 0.9998 when n= 15 and 20, respectively. P1(n, .95, .95) =
0.920,0.979 and 0.995 when n= 15,20 and 25, respectively.
Recall that the two-sided TI is the interval in (3) with kreplaced by kt, where kt
is determined by Eqn (9). We computed P2(n, p, γ) = P(tn−1(3.3×√n)>√nkt)<
P(¯
X1/S1> kt) along the lines of the preceding paragraphs as follows. P2(n, .90, .95) =
0.941,0.988 and 0.998 when n= 15,20 and 30, respectively. P2(n, .95, .95) = 0.941,0.963,
0.985 and 0.994 when n= 15,20,30 and 40, respectively. The equal-tailed TI in
(3) with kreplaced by ke, where keis determined by Eqn (10). We computed
P3(n, p, γ) = P(tn−1(3.3×√n)>√nke)< P (¯
X1/S1> ke) for some values of (n, p, γ)
as follows. P3(n, .90, .95) = 0.841,0.948 and 0.996 when n= 15,20 and 30, respec-
tively. P3(n, .95, .95) = 0.594,0.755,0.927 and 0.982 when n= 15,20,30 and 40,
respectively.
