In this paper, a study on the length of confidence intervals of the mean
in a Normal population when the variance is either known or unknown is
carried out. It is also proved that ignoring the population variance always
leads to a worse interval than when such a variance is taken into accoun

González Abril, Luis

Sánchez-Reyes Fernández, Luis María

Velasco Morente, Francisco

idUS. Depósito de Investigación Universidad de Sevilla

Bolet´ın de Estad´ıstica e Investigacio´n OperativaVol. 26, No. 1, Febrero 2010, pp. 19-24Estad´ısticaSome commentaries on confidence intervals of the mean ina Normal populationL. Gonza´lez Abril, L.M. Sa´nchez-Reyes and F. Velasco MorenteDepartamento de Economı´a Aplicada IUniversidad de SevillaB luisgon@us.es, luiss-rf@us.es, velasco@us.esAbstractIn this paper, a study on the length of confidence intervals of the meanin a Normal population when the variance is either known or unknown iscarried out. It is also proved that ignoring the population variance alwaysleads to a worse interval than when such a variance is taken into account.Keywords: Student‘s t population, Chi-squared population.AMS Subject classifications: 62F99.1. IntroductionLet X1, X2, · · · , Xn be independent and identically distributed (i.i.d.) ran-dom variables (a sample) from an N(µ, σ) population, and let α be a specifiedvalue such that 0 < α < 1. If σ is known then the shortest length 1 − α con-fidence interval based on the pivotal quantity1 (pivot) X−µσ/√n∼ N(0, 1) of themean µ is (X − 1√nσ z1−α/2 , X +1√nσ z1−α/2)(1.1)where X is the mean of the sample and z1−α/2 is the (1−α/2)th quantile of theN(0, 1) population. If σ is unknown then the shortest length 1 − α confidenceinterval based on the pivot X−µS/√n−1 ∼ tn−1 of the mean µ is(X − S√n− 1 tn−1,1−α/2, X +S√n− 1 tn−1,1−α/2)(1.2)where S is the standard deviation of the sample and tn−1,1−α/2 is the (1−α/2)thquantile of the Student’s t population with n − 1 degrees of freedom (see forinstance [1]).1The symbol “∼” must be read as “distributed as”.c© 2010 SEIO20 L. Gonza´lez Abril, L.M. Sa´nchez-Reyes, F. Velasco MorenteLet us consider a toy example: suppose that x1 = 0.95 and x2 = 1.05is a sample from an N(µ, σ) population and two researchers want to builda 0.90 confidence interval of the unknown parameter µ. The first researcherknows that σ = 1 and therefore, from (1.1), the confidence interval of µ isI1 = (−1.630 , 2.163). But for the second researcher, σ is unknown and there-fore, from (1.2), the confidence interval will be I2 = (−0.684 , 1.316). In viewof these intervals the following question arises: how can I2 be more accurate(shorter) than I1 when the latter needs less information? The main aim of thispaper is to answer this question.2. Ratio of lengthsIntervals (1.1) and (1.2) have the same center (X) although different radii.The length of the interval (1.1) is 2√nσ z1−α/2, which only depends on the sizeof the sample, whereas the length of the interval (1.2) is 2√n−1S tn−1, 1−α/2, thatdepends on the sample because S = S(X1, · · · , Xn) is a statistic. To comparethese two lengths, we are going to study the random ratio between them whichwill be denoted by C = C(X1, · · · , Xn), that is, C = tn−1,1−α/2z1−α/2√n√n−1Sσ .Let us recall two well-known results (see for instance [1]): i) If X1, · · · , Xnare i.i.d. random variables from an N(µ, σ) population, then nS2σ2 ∼ χ2n−1; andii) if Y ∼ χ2p then E[Y r] = 2rΓ(r+ p2 )Γ( p2 ), where Γ(x) is the gamma function and rand p are positive. Hence, the probability density function (pdf) of C, denotedby fC , is given asfC(c) =(z1−α/2tn−1,1−α/2)n−1(n−1)n−122n−32 Γ((n−1)/2)cn−2exp(−n−12(z1−α/2tn−1,1−α/2c)2), c > 0.(2.1)This function is depicted in Figure 1, and it can be seen that its maximum isnear to 1 and the tails are tighter as n increases. It is also straightforward tocalculate the expectation µc = E[C] and the variance σ2c = V ar[C]:µc =tn−1,1−α/2z1−α/2√2 Γ(n/2)√n− 1 Γ((n− 1)/2) , andσ2c =(tn−1,1−α/2z1−α/2)21−( √2 Γ(n/2)√n− 1 Γ((n− 1)/2))2 .Furthermore, if n is large then2 (see for instance [3]) tn , p ≈ zp for any0 < p < 1 and limx→+∞Γ(x+1/2)√x Γ(x)= 1 (see for instance [4]), and hence2The symbol ≈ must be read as “approximately equal to”.Some commentaries on confidence intervals of the mean in a Normal population 21Figure 1: Some pdfs of C with α = 0.05.limn→+∞√2 Γ(n2 )√n−1 Γ(n−12 )= 1. Therefore limn→+∞µc = 1 and limn→+∞σ2c = 0, which implythe convergence in probability of the random variable C to 1 (see for instance[2]), that is, the intervals (1.1) and (1.2) are asymptotically equivalent.But if n is not very large then these two intervals can widely differ. To showit, Table 1 and Table 2 have been built for several values of α, n and k. So forTable 1: Values of µc and σc for some n and α.α = 0.10 α = 0.05 α = 0.01n µc σc µc σc µc σc2 3.0627 2.3139 5.1726 3.9079 19.7179 14.89713 1.5733 0.8224 1.9455 1.0170 3.4147 1.78504 1.3182 0.5563 1.4960 0.6313 2.0891 0.88165 1.2183 0.4422 1.3316 0.4834 1.6801 0.609910 1.0840 0.2588 1.1226 0.2680 1.2272 0.293025 1.0294 0.1493 1.0421 0.1512 1.0746 0.1559100 1.0069 0.0716 1.0098 0.0719 1.0171 0.0724500 1.0014 0.0317 1.0019 0.0317 1.0033 0.0318instance, Table 2 shows that the event considered in the toy example, that is,that the interval (1.2) is shorter than the interval (1.1) for a sample with n = 2and α = 0.10, has a probability equal to 0.2055.Table 2: Values of P (C < k) for some n, k and α.α = 0.10 α = 0.05 α = 0.01n k=0.75 k=0.9 k=1 k=0.75 k=0.9 k=1 k=0.75 k=0.9 k=12 0.1549 0.1854 0.2055 0.0921 0.1104 0.1226 0.0242 0.0291 0.03233 0.1635 0.2267 0.2719 0.1102 0.1547 0.1874 0.0372 0.0531 0.06514 0.1564 0.2439 0.3098 0.1128 0.1798 0.2321 0.0454 0.0751 0.09985 0.1454 0.2511 0.3340 0.1091 0.1938 0.2630 0.0492 0.0924 0.130510 0.0937 0.2471 0.3885 0.0759 0.2087 0.3375 0.0433 0.1307 0.226025 0.0260 0.1955 0.4317 0.0221 0.1749 0.3995 0.0146 0.1303 0.3236100 0.0001 0.0659 0.4664 0.0001 0.0612 0.4503 0.0001 0.0507 0.4113500 0.0000 0.0006 0.4850 0.0000 0.0005 0.4779 0.0000 0.0005 0.460322 L. Gonza´lez Abril, L.M. Sa´nchez-Reyes, F. Velasco Morente3. Correction of confidenceIt still remains the previous question: if σ is known and the interval (1.2) isshorter than the interval (1.1), would it be better to choose the shortest one asthe confidence interval of µ? The answer is negative. For such a purpose we areproving that if C = c is known then the confidence coefficient of the interval (1.2)is not longer 1− α but 2Φ(z1−α/2 c)− 1, where Φ is the cumulative distributionfunction (cdf) of the N(0, 1) distribution, value which is smaller than 1−α whenc < 1.To see this, we begin by noting that the interval (1.2) will not cover the trueparameter µ just if the following event takes place{∣∣X − µ∣∣ ≥ 1√n− 1S tn−1 , 1−α/2}.Denoting Z =√n(X−µ)σ , this event can be written as {|Z| ≥ z1−α/2 c}. Itsprobability depends on c and shall be written as:p(c) = P{|Z| ≥ z1−α/2 c}. (3.1)Taking into account that Z ∼ N(0, 1) no matters the value of c (since X and Sare independent), the expression on the right-hand side of the inequality of theevent given in (3.1) for each fixed value of c will be the (1 − p(c)/2)th quantileof the N(0, 1) distribution, that isz1−p(c)/2 = z1−α/2 c. (3.2)By applying the transformation Φ in (3.2) then:Φ(z1−p(c)/2) = Φ(z1−α/2 c)⇒ 1− p(c)/2 = Φ(z1−α/2 c)since Φ(zp) = p, and hence1− p(c) = 2Φ(z1−α/2 c)− 1. (3.3)That is, if σ is known then 2Φ(z1−α/2 c)− 1 is the true confidence of the interval(1.2) as we wanted to prove. Furthermore, as Φ is a strictly increasing function(since it is the cdf of the N(0,1) distribution) then 1−p(c) is a strictly increasingfunction too, and hence, if c < 1 then1− p(c) < 1− p(1) = 2Φ(z1−α/2)− 1 = 2(1− α/2)− 1 = 1− α.But as c can be any positive number, the corrected confidence of the interval(1.2) can be also considered a random variable 1 − p(C), and it can be worthSome commentaries on confidence intervals of the mean in a Normal population 23noting that its expectation is 1−α in coherence with the initial confidence coef-ficient. To see it, let φ and fC the pdfs of Z and C respectively. Thus, φ(z)fC(c)is the joint pdf of the random vector (Z,C), since both random variables areindependent, and thereforeE[1− p(C)] =∫ ∞0P (|Z| < z1−α/2c) fC(c) dc=∫ ∞0(∫ z1−α/2c−z1−α/2cφ(z) dz)fC(c) dc= P(|Z| < z1−α/2C) = P(∣∣X − µ∣∣S√n− 1 < tn−1,1−α/2)= 1− α.Considering still C as a random variable, let us study how often the confidence1 − p(C) of the random interval (1.2) will be considerably smaller than 1 − α,and to this end, we are going to evaluate the probability of the{1−p(C)1−α < k}event for several values of k. It follows from (3.3) thatP(1− p(C)1− α < k)= P(C <Φ−1 ((1 + k(1− α))/2)z1−α/2)and from this result, Table 3 has been obtained for different values of n, k andα. This table suggests that the probability of a substantial correction decreasesTable 3: Values of P(1−p(C)1−α < k)for some n, k and α.α = 0.10 α = 0.05 α = 0.01n k=0.75 k=0.9 k=0.95 k=0.75 k=0.9 k=0.95 k=0.75 k=0.9 k=0.952 0.1239 0.1644 0.1826 0.0667 0.0913 0.1038 0.0142 0.0201 0.02363 0.1074 0.1825 0.2205 0.0593 0.1084 0.1379 0.0129 0.0257 0.03544 0.0866 0.1819 0.2351 0.0467 0.1103 0.1537 0.0097 0.0267 0.04235 0.0687 0.1754 0.2402 0.0355 0.1061 0.1601 0.0068 0.0250 0.045010 0.0218 0.1323 0.2294 0.0084 0.0721 0.1511 0.0008 0.0119 0.036825 0.0009 0.0552 0.1696 0.0001 0.0197 0.0938 0.0000 0.0008 0.0103100 0.0000 0.0012 0.0436 0.0000 0.0001 0.0096 0.0000 0.0000 0.0000500 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000when α tends to zero and n tends to infinity, but however when n is not very largethen there is a reasonably high probability of a certain loss in the re-evaluatedconfidence. A significant correction of the confidence is more likely to happenwhen the initial confidence is not high.In order to complete this study, let us obtain the pdf of the corrected confi-dence, which is denoted by fCo. From (2.1), it can be written that for 0 < u < 1:fCo(u) =(Φ−1)′(u+12 )2z1−α/2fC(Φ−1(u+12 )z1−α/2)24 L. Gonza´lez Abril, L.M. Sa´nchez-Reyes, F. Velasco Morentewhere(Φ−1)′denotes the derivative of Φ−1, and since(Φ−1)′(u)=√2pi exp(z2u/2),then by substitution,fCo(u) =(n−1)n−12 √pi2n2−1 Γ((n−1)/2)(tn−1,1−α/2)n−1(z(u+1)/2)n−2exp(12(z(u+1)/2)2)exp(− (n−1)(z(u+1)/2)22(tn−1,1−α/2)2).ConclusionsIn this paper it has been shown that ignoring σ leads always to a worseconfidence interval because the interval (1.2) will be longer in general, but ifeven it were shorter then the gained accuracy is false since there is a loss of thereal confidence.References[1] Casella, G. and R. Berger (1990). Statistical Inference. Duxbury Press.Belmont, California (USA).[2] Chung, K. L. (1974). A Course in Probability Theory (second ed.). NewYork: Academic Press.[3] Koehler, K. (1983). A simple approximation for the percentiles of the tdistribution. Technometrics, 25(1), 103–105.[4] Rudin, W. (1974). Real and Complex Analysis. Mc Graw-Hill, Inc. NewYork.About the authorsLuis Gonza´lez Abril is a Lecturer in Department of Applied Economics I inthe University of Seville. He obtained the M.Sc. degree in Mathematics and thePh.D. degree in Economics from the University of Seville. His research interestsinclude machine learning, statistical methodology and classification methods.Luis Mar´ıa Sa´nchez Reyes is a Lecturer in Department of Applied EconomicsI in the University of Seville. He obtained the M.Sc. degree in Mathematics fromthe University of Seville. His research interests include statistical methodologyand classification methods.Francisco Velasco Morente is a Lecturer in Department of Applied EconomicsI in the University of Seville. He obtained the M.Sc. degree and the Ph.D. degreein Mathematics from the University of Seville. His research interests includedynamical systems, statistical methodology and classification methods.

Some commentaries on confidence intervals of the mean in a Normal population

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Some commentaries on confidence intervals of the mean in a Normal population

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