Sample Size for a Specified Width Confidence Interval on the Ratio of Variances from Two Independent...

Sample Size for a Specified Width Confidence Interval on the Ratio of Variances from TwoIndependent Normal PopulationsAuthor(s): Aaron GoldmanSource: Biometrics, Vol. 19, No. 3 (Sep., 1963), pp. 465-477Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2527935 .

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SAMPLE SIZE FOR A SPECIFIED WIDTH CONFIDENCE INTERVAL ON THE RATIO OF VARIANCES FROM

TWO INDEPENDENT NORMAL POPULATIONS'

AARON GOLDMAN

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico, U. S. A.

1. INTRODUCTION

The experimenter who desires an interval estimate of a parameter should take into consideration two very fundamental questions: (1) Does the interval contain the parameter? (2) Is the interval too wide? In practice these questions may be answered in the form of probability statements. Let the confidence coefficient, 1 - a, denote the probability the interval will contain the parameter and let the width coefficient, O3, denote the probability that w, the width of the interval, will be less then a specified number, d. Usually 1 - a is specified, a confidence interval is computed, and the width is then governed greatly by the sample size, n. The larger n becomes, the more precise the estimator becomes; hence a smaller interval. Consequently, the larger the sample size, the larger 32 becomes. The crux of the problem may now be stated: Determine the sample size, n, necessary for (A) a 1 - a confidence interval and (B) a 32 width coefficient. In order to provide an answer some a priori information is needed. This suggests a two step procedure.

The purpose of this paper is to utilize a two step method to determine the size of the samples from two independent normal populations necessary to satisfy (A) and (B) when obtaining an interval estimate on the ratio of variances. The technique requires taking preliminary samples of sizes ml and mi2, making a simple calculation after specifying a, A, and d, and looking for the sizes of n, and n? in Table I to insure obtaining the desired results. Table I gives choices of n, and n2 for I - a and 3 = .90, .95, and .99.

It is of importance to note that there is not a unique solution for n1 and n, but in general n, + n2 can be minimized without much difficulty.2 Also it is important to bear in mind that the width co-

'A part of this paper was written while the author was sponsored by a National Science Foundation fellowship at Oklahoma State University.

21f there is a cost involved then the total cost, C = cini + C2f2, needs to be minimized. (See example 2).

465



466 BIOMETRICS, SEPTEMBER 1963

efficient, 12, is the greatest lower bouiid oln the probability that w is less than d. The upper boulid is given as 2,3 - 32. Thus in Table I which lists n* and n* for 13 the bounds are as follows:

( Lower Bound Upper Bound .90 .8100 .9900 .95 .9025 .9975 .99 .9801 .9999

The proof appears in Theorem 2, Section I of the Appendix. The theory for this procedure is given by Graybill [1958] and follows along the line of the work done by Graybill and Morrison [1960]. Examples are given to demonstrate the technique and the required theory is presented in the Appendix. Section I of the Appendix is devoted to the general technique, Section II to the specific application, and Section III to a one step technique.

2. EXAMPLES

Example 1

Suppose a 1 - a confidence interval is desired on the ratio of variances, 0 = -2//_2 , so that the probability that the interval width is less than d lies between 12 and 23 - 12. Suppose a is set equal to .05 and 13 is .90. Because it is known that interval estimates of ratios of variances are in general quite wide, d is set equal to 2. Preliminary samples of sizes in1 = 3 and m2 = 10 are taken and sample variances are computed in the usual way, i.e.

in 1 712

82 (Xi ;0)1) M Z (y 1)2 s1- E (xi - and s2 /(m2 i=l i=l

where

x = ZxXi/m1 and Y= Zyj/m2.

Suppose a value of z = s2/s2 = .166 is computed. The quantity Q = dF#(m2 , mjn/z, where FB(m2 , Mnj) is the fractile of the cumulative F distribution with m2 - 1 and m1 - 1 degrees of freedom such that the probability of a larger F value is 13, is computed. The values of Q are the given entries in Table I. Appropriate values of n* = n- 1 and n* = n2- 1 are selected. In this case

Q = 2 X .332/.166 = 4,

and the following choices of n* and n* are available:



A SPECIFIED WIDTH CONFIDENCE INTERVAL 467

Choice 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 n* 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 120 n* 120 55 35 29 26 23 21 19 18 17 15 13 12 11 11 10

Thus the total sample size is miniimized for ni = 19, n2 = 20; n1 = 20, n2 = 19; and n, = 21, n2, = 18.

It is interesting to note that had a confidence interval been placed on 0 after the first m1 and m2 observations were taken, the interval width would have been 6.511.

This example is applicable in Analysis of Variance problems. If a specified width interval is desired when placing a confidence interval on the ratio of two independent mean squares in an Analysis of Vari- ance, then in order to determine the necessary sample sizes the steps in Example 1 are followed.

A detailed general application of the method is presented in Section II of the Appendix.

.Example 2

When a cost is involved it is desirable to minimize

C = clnl + C2n2

where cl is the cost of taking one of the n, samples and c2 is the cost of taking one of the n2 samples. If in Example 1 the costs were cl = $2.00 and c2 = $1.00, then the following results would be obtained (the choices of ni and n2 are the same as in Example 1):

Choice 1 2 3 4 5 6 7 8 9 10 Cost $145 82 64 60 59 58 58 58 59 60

Thus the cost is minimized for n1 = 17, n2 = 24; ni = 18, n? = 22; and n1 = 19, n2 = 20.

Example 3

The tables might be used in a somewhat different manner. If values of n1 and n2 are wanted so that w, the width of the interval on 0, is less than lOOp percent of 0 with width coefficient ,B, solutions are found in only one step. Note that the width coefficient is precisely ,B and d = pG. Thus P[w < pO] > /3. A complete proof of this technique is given in Section III of the Appendix.

Suppose a 95 percent confidence interval to be placed on 0 = S2 1

with B = .95, and width equal to 300 percent of 0. (Note that smaller percentages are not practical, because of the fact that in general small samples do not provide a precise estimate of 0. For instance, if p were equal to .10 then the necessary sample sizes would be too large




for practical purposes. Table I does not include any percentages less than 100 because of the large sample sizes required.) Table I provides the necessary sample sizes. Values of p are entries in the Table and corresponding values of n* and n* are then chosen. In this example the following results may be obtained:

Choice 1 2 3 4 5 6 7 8 n* 18 19 20 21 25 30 29 40 n* 102 80 60 50 38 28 30 22

Thus samples of sizes ni = 31 and n2 = 29 minimize the total sample size and P[w < 30] 2 .95.

3. APPENDIX

Section 1: Necessary theorems

Theorems that are applied are given as follows :3

THEOREM 1: Let the chance variable w be the width of a confidence interval on a parameter 0 based on samples of sizes n, and n2 . Suppose that w depends on n, and n2 and on an unknown parameter 0* (0* may be the parameter 0). Suppose also there exists a function of w, 0*, ni and n2, say g(w; 0*, ni , n2) such that if Y = g(w; 0*, ni , n2) then the distribution of Y does not depend on any unknown parameters except ni and n2 . Let f(n1 , n2) define a function such that

P[Y < f(n, , n2)]= 3 for any 0 < 3 < 1.

Let the solution of the equation g(w; 0*, ni , n2) = f(n, , n2) for w be w = h(6*, n1, n2), and suppose the following are true for w > 0:

(a) g(w; 0*, ni, n2) is monotonic increasing in w for every n1, n2 and 0.

(b) h(0*, n1, n2) is monotonic increasing for every ni and n2, and monotonic decreasing in ni and n2 for every 0*.

(c) z is a random variable which is available from step one of the procedure such that P[t(z) > 0*] = ,B for 0 < 13 < 1 where t(z) is a function of z which does not depend on any unknown parameters or on ni and n2 .

Let d and 13 be specified in advance. Then if n1 and n2 are such that the equation

h[t(z);ni ,n2] < d

is satisfied [t(z) is known] then the following inequality is true:

3Theorem 1 is stated precisely as by Graybill [1958] with slight changes in notation.




P[w < d] > 02.

THEOREM 2: Given the same conditions as those stated in Theorem 1 then

P[w < d] < 23 - 12

Proof: From Graybill [1958],

P[w < d] = Pi + P2

where

Pi = P(w < h[t(z),n] 1t(z) < 6)P[t(z) < 0]

p2 = P(w < h[t(z), n] 1t(z) ? 6)P[t(z) 2 0]

d = h[t(z), n]

P[t(z) < 0] = 1-

and

P[t(z) 2 6] = 1

As t(z) c-o

P(w < h[t(z),n] It(z) < 0,t(z) o) - 0.

This follows from the assumption that h[t(z), n] is monotollically increasing for every n.

As t(z) -> 0 from either direction

P(w < h[t(z), n] I t(z) -60) -> P[w < h(6; n)] = .

If t(z) > 0 and t(z) -> o then

P(w < h[t(z), n] I t(z) > 0, t(z) - ) > 1.

The maximum of Pi + P2 can now be obtained. Since P[t(z) <6] =

1 - 1 and P(w < h[t(z), n] I t(z) < 0) obtain simultaneous maximums as t(z) -* 0, it follows

Max {p,} = 1(1 - -3).

Similarly P2 is maximized simultaneously with P(w < h[t(z), n] I t(z) > 0). As t(z) -> co and since P[t(z) > 0] 13, it follows

Max {p2} =.

Hence

Max {Pi + P21 203 _ 32.




It is important to keep iii mind that although the sizes of the samliples that are drawn in the first step, mn1 and n,2 , are fixed, the sizes of the samples drawn in the second step, n1 and n2 , are random variables. They are a function of the random variable, z, and this fact is of para- mount importance in the entire procedure.

Section II: Application

An application of the theorems is presented. First of all, samples of sizes m1 and mn2 are drawn from two independently distributed normal populations, 7r1 and 7r2 . The quantity z is computed. Then n1 and n2 are chosen by a simple computation and use of Table I. The resultant 1 - a confidence interval on the ratio of variances, detailed computations, and use of the table will follow.

A 1- a interval on the ratio of variances is given by

P[s2Fl(a/2)(nl, n2)/sl < < s82Fa/2(nl, n2)/sl] = 1-o.

where L=2/1 2

nl n2 2 = Z (Z -iz)2/n1* 2 s = (z2-)2/*

n* = n-i, n2* = n2 - ,

ni n2

Z1 = Zz1i/nl , Z2 = ZZ2j/n2

z1 is l distributed normally with mean u1 and variance o-1, Zii is distributed normally with mean ul and variance l2 Z2j iS distributed normally with mean U2 and variance aT2,

ni is the size of the sample drawn from 7r1, n2 is the size of the sample drawn from 7r2,

and F,(n1 , n2) is such that

fX7(nln2 W(F;ni ,n2) dF = -y ,Y (nl ,n2)

where W is the F distribution with n*j and n* degrees of freedom. The following equations may be shown to satisfy the conditions

of the theorem:

(a) g(w; 0*, n1 , n2) = w/OC(n1 , n2)

where

C(nl , n2) = F./2(nl , n2) - (1/Fa/2(n2 , n1)) = 1

(b) h(6; ni, n2) = OC(n, ,n2)Fl-p(n2, n1).




(e) z = (nl -1) (Z2- )2/(M 1) E (Zl2 - *) q-1 V=1

where ml I2

z= z1jp/m and Z2* Z2a/m2

As a result the smallest integral values of n1 and n2 are chosen so that

Fj-ft(n2 I njQCnl, In2) < dF(M2 I Mj)/Z.

Table I provides values of n* and n* so that the only calculation necessary is

Q'M . "M -= dF#(m2 , ml)/Z.

Section III: A one step technique

The tables might be used in a somewhat different manner. If the experimenter wants values of n1 and n2 so that the width of the interval on the ratio of variances, 0 = 02/a2, is less then p percent of 0 with width coefficient 3, solutions for n1 and n2 are found in but one step.

The problem may be stated as follows: Find n1 and n2 so that

P[w < p6/100] -,.

This may be written as

P{s2[Fai2(nl , n2) - (1/Fai2(n2 , nj)]/sl ? po2/1OOo2} ?2

or

P 281 <- p/100[Fa,2(nl , n2) - ( /Fa/2(n2 , n,))]A} I A.

It is known that

2 1 1sI < F1%(n2 , nj)] = A .

Thus the solution is found by choosing the smallest integral values of n, and n2 so that

p/100C(n1 , n2) 2 Fl-p(n2, n1)

or

p/iOO > Flf(n2 , nl)C(n, , n2)

or

p/100 2 Qm>,m.

The entries in Table I provide values of Q.




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ACKNOWLEDGMENTS

The author is indebted to Dr. Franklin A. Graybill for his guidance in this work.

REFERENCES

Graybill, F. A. [195S]. Determining Sample Size for a Specified Width Confidence Interval. The Annals of Mathematical Statistics 29, 282-87.

Graybill, F. A. and Morrison, R. D. [1960]. Sample Size for a Specified Width Confidence Interval on the Variance of a Normal Distribution. Biometrics 16, 636-41.

Goldman, A. [1961]. On the Determination of Sample Size. Los Alamos Scientific Laboratory Report LA-2520.



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