Testing manufacturing performance based on capability index C-pm

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DOI 10.1007/s00170-004-2182-8

O R I G I N A L A R T I C L E

P.C. Lin · W.L. Pearn

Testing manufacturing performance based on capability index C

_pm

Received: 15 December 2003 / Accepted: 19 March 2004 / Published online: 10 August 2005 ©Springer-Verlag London Limited 2005

Abstract The loss-based process capability index Cpm, some-times called the Taguchi index, has been proposed to the manu-facturing industry to measure process performance. In this paper, we propose a method to assess the performance of a normally distributed process. We implement the theory of testing hypoth-esis using the natural estimator of Cpm, and provide an efficient program to calculate the p-values. We also provide tables of the critical values for some commonly used capability requirements. Based on the test, we develop a simple step-by-step procedure for in-plant applications.

Keywords p-value· Process capability index · Taguchi index · Testing hypothesis

1 Introduction

Process capability indices (PCIs) provide numerical measures on whether a manufacturing process meets the preset quality require-ment. We note that PCIs have been the focus of recent research in quality assurance and process capability analysis. Three basic ca-pability indices Cp, Cpk, and Cpm, which establish the relationship between the actual process performance and the manufacturing specifications, have been widely used in the manufacturing indus-try. These indices are explicitly defined as (see [1, 2]):

Cp=USL− LSL 6σ , Cpk= min _USL_{− µ} 3σ , µ − LSL 3σ , P.C. Lin (u)

Center of General Education,

National Chin-Yi Institute of Technology, Taiwan, R.O.C.

E-mail: linpc@ncit.edu.tw Tel.: +886-4-2324505 ext. 2254 W.L. Pearn

Department of Industrial Engineering & Management, National Chiao Tung University,

Taiwan, R.O.C.

Cpm=

USL− LSL 6σ2_{+ (µ − T)}2,

where USL is the upper specification limit, LSL is the lower specification limit,µ is the process mean and σ is the process standard deviation, and T is the target value preset by the prod-uct designer or manufacturing engineer. We will focus on the situation in which the specification interval is two-sided with the target value T at m= (USL +LSL)/2, the midpoint of the speci-fication interval (LSL, USL), which is most common in practice. The index Cp measures the consistency of process quality characteristic relative to the manufacturing tolerance and, there-fore, only reflects process potential (or process precision). The index Cpk takes into account the magnitudes of process vari-ation as well as the degree of process centering, which mea-sures manufacturing performance based on yield (proportion of conformities). Hence, the capability index Cpkis a yield-based index.

The index Cpmemphasizes measuring the ability of the pro-cess to cluster around the target, which therefore reflects the degrees of process targeting (centering). The index Cpm incor-porates with the variation of production items with respect to the target value and the specification limits preset in the fac-tory (see [3–5]). It is noted in some recent quality research and capability analysis works, that both of the Cpkand Cpm in-dices provide the same lower bounds on process yield, that is, Yield≥ 2Φ(3Cpk) − 1 = 2Φ(3Cpm) − 1. Pearn and Lin [6] in-vestigated the behavior of the actual process yield for processes with fixed index value of Cpk= Cpm, but with different degrees of process centering. The results in Pearn and Lin [6] illustrate the advantage of using the index Cpm over the index Cpk in measuring manufacturing performance, as Cpm provides better protection to the customers.

In fact, the capability index Cpmis not primarily designed to provide an exact measure on the number of conforming items, i.e., the process yield. But Cpmconsiders the process departure (µ − T)2_{(rather than 6}_{σ alone) in the denominator of the} defin-ition to reflect the degrees of process targeting (see [2, 3]). We note thatσ2+ (µ − T)2= E[(X − T)2], which is the major part

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of the denominator of Cpm. Since E[(X − T)2] is the expected

loss where we note that the loss of a characteristic X missing the target is often assumed to be well approximated by the symmet-ric squared error loss function, loss(X) = (X − T)2. Hence, the capability index Cpmis a loss-based index.

In general the process meanµ and the process variance σ2 are unknown. But, in practiceµ and σ2can be estimated using the sample data. We then consider the estimated index ˆCpmthe

natural estimator of the index Cpm. In order to calculate the

es-timator, however, sample data must be collected, and a great degree of uncertainty may be introduced into capability assess-ments owing to the sampling errors. The approach by simply looking at the calculated values of the estimated indices and then make a conclusion on whether the given process is capable, is highly unreliable as the sampling errors have been ignored. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estima-tors and their sampling distributions on constructing confidence intervals and performing hypothesis testing. For normally dis-tributed processes, Cheng [7] has developed a hypothesis testing procedure where tables of the approximate p-values were pro-vided for some commonly used capability requirements, using the natural estimator of Cpm. The practitioners can use the

ob-tained results to determining if their process satisfies the targeted quality condition. But Cheng’s approach requires further estima-tion of the distribuestima-tion characteristic(µ−T)/σ when calculating the p-values, which introduces additional sampling errors thus making the decisions less reliable.

In this paper, we first obtain an explicit form of the cumulated distribution function of the maximal likelihood estimator (MLE) of Cpm, which can be expressed in terms of a mixture of the

chi-square distribution and the normal distribution. We then im-plement the theory of testing hypothesis using the MLE of Cpm,

and provide an efficient Maple program to calculate the p-values. To eliminate the need for estimating the distribution characteris-tic, the parameterξ = (µ − T)/σ, we examine the behaviors of the p-values as well as the critical values c0against the

param-eterξ. Then, we develop a simple step-by-step procedure where tables of critical values are provided for some commonly used capability requirements. The practitioners can also use the pro-posed procedure, without having to run the Maple program, to determine whether their manufacturing process meets the preset capability requirement, and make reliable decisions.

2 Distribution of the estimated C

pm

The index Cpmcan be rewritten as the following:

Cpm=

d

3σ2_{+ (µ − T)}2,

where d= (USL − LSL)/2 is half of the length of the specifi-cation interval. In generally, both the process mean µ and the process varianceσ2are unknown, as pointed out earlier. The es-timated index ˆCpmis obtained by replacingµ and σ2 by their

estimators. Chan et al. [2] and Boyles [8] proposed two different estimators of Cpmrespectively defined as the following:

ˆCpm(CCS)= d 3 S2_{+ ( ¯X − T)}2 and ˆCpm(B)= d 3 S2 n+ ( ¯X− T)2 , where ¯X=n_i₌₁Xi/n, S2=ni=1(Xi− ¯X)2/(n − 1) and S2n= n

i=1(Xi− ¯X)2/n. In fact, the two estimators, ˆCpm(CCS) and

ˆCpm(B), are asymptotical equivalent. We note that ¯X and S2n are

the MLEs ofµ and σ2, respectively. Hence, the estimated index ˆCpm(B)is the MLE of Cpm. Furthermore, the term S2n+ ( ¯X − T)2

in the denominator of ˆCpm(B)is the uniformly minimum variance

unbiased estimator (UMVUE) of the termσ2+ (µ − T)2 in the denominator of Cpm, where S2n+ ( ¯X − T)2=

n

i=1(Xi− T)2/n

andσ2+ (µ − T)2= E[(X − T)2]. Therefore, it is reasonable for reliability purpose, that we adapt the estimator ˆCpm(B)to evaluate

the performances of normally distributed processes in this paper and define the estimated index ˆCpm= ˆCpm(B).

Under the assumption of normality, Kotz and Johnson [4] ob-tained the rth moment, and calculated the first two moments, the mean, and the variance of ˆCpm. Zimmer and Hubele [9] provided

tables of exact percentiles for the sampling distribution of the es-timator ˆCpm. Zimmer et al. [10] proposed a graphical procedure

to obtain exact confidence intervals for Cpm, where the

parame-ter (µ−T)/σ is assumed to be a known constant, thus the results obtained is somewhat unreliable. On the other hand, using the method similar to that presented in Vännman [11], we may ob-tain an exact form of the cumulative distribution function of ˆCpm.

Under the assumption of normality, the cumulative distribution function of ˆCpmcan be expressed in terms of a mixture of the

chi-square distribution and the normal distribution: F_ˆC pm(x)= 1− b√n/(3x) 0 G b2n 9x2− t 2 φ(t+ ξ√n) + φ(t − ξ√n)dt, (1) for x> 0, where b = d/σ, ξ = (µ − T)/σ, G(·) is the cumula-tive distribution function of the chi-square distributionχ_n2₋₁, and

φ(·) is the probability density function of the standard normal

distribution N(0, 1).

3 Testing process performance

To test whether a given process is capable, we may consider the following statistical testing hypotheses:

H0: Cpm≤ C (process is not capable),

H1: Cpm> C (process is capable).

Based on a givenα(c0) = α, the chance of incorrectly concluding

an incapable process (Cpm≤ C) as capable (Cpm> C), the

deci-sion rule is to reject H0(Cpm≤ C) if ˆCpm> c0and fails to reject

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For processes with target value setting on the middle of the specification limits (T= m = (USL + LSL)/2), which are fairly common situations, the index may be rewritten as: Cpm=

b/[3(1 + ξ2)1/2]. Given Cpm= C, b = d/σ can be expressed as

b= 3C(1 + ξ2)1/2. Given a value of C (the capability require-ment), the p-value corresponding to c∗, a specific value of ˆCpm

calculated from the sample data, is (by Eq. 1): P( ˆCpm≥ c∗|Cpm= C) = b√n/(3c∗) 0 G b2n 9(c∗)2− t 2 _{φ(t + ξ}√_n_{) + φ(t− ξ}√_n₎_dt_{. (2)}

Given values ofα and C, the critical value c0can be obtained

by solving the equation P( ˆCpm≥ c0|Cpm= C) = α. Hence, given

values of capability requirement C, parameterξ, sample size n, and riskα, the critical value c0 can be obtained by solving the

following equation: b√n/(3c0) 0 G b2n 9c2₀− t 2 φ(t + ξ√n) + φ(t − ξ√n)dt= α . (3) It is noted that Eq. 2 is an even function ofξ for fixed values of c∗, n and C. Therefore, given values of n and C the p-values corresponding to a specific value c∗forξ = ξ0 andξ = −ξ0are

the same. Similarly, given values of C, n andα the critical values c0forξ = ξ0andξ = −ξ0are the same.

An efficient Maple program is developed for calculating Eq. 2 to obtain the p-value for given c∗. We note that similar programs can also be written using software “Mathematica” or “MatLab”. The program is listed below, with input parameters set to LSL= 2.40, USL = 3.40, T = 2.90, C = 1.00, n = 100, ¯X = 2.825, Sn= 0.125 as an example. Here, we set ξ = ˆξ = ( ¯X−

T)/Sn, since generally ξ = (µ − T)/σ is unknown. This

ap-proach is similar to one proposed by Cheng [7]. On the other hand, c∗= ˆCpm can be calculated from the sample data. The

program gives ˆξ = −0.6, ˆCpm= 1.1433, and the corresponding

p-value as 0.0338.

Maple program:

> # input parameter values LSL, USL, T, C, n, X_bar, Sn.

LSL:=2.40; USL:=3.40; T:=2.90; C:=1.00; n:=100; X_bar:=2.825; Sn:=0.125;

d:=(USL - LSL)/2;ξ:=(X_bar - T)/Sn; c1:=d/(3∗(Snˆ2 + (X_bar - T)ˆ2)ˆ0.5): # Note that c1 = c∗= Cpm_hat. b:=3∗C∗(1 + ξˆ2)ˆ0.5: G:=(c1,t)->stats[statevalf,cdf,chisquare[n - 1]] ((bˆ2∗n)/(9∗c1ˆ2) - tˆ2): h:=t->stats[statevalf,pdf,normald](t + ξ∗nˆ0.5) + stats[statevalf,pdf,normald](t -ξ∗nˆ0.5): pV:=c1->int(G(c1,t)∗h(t),t=0..(b∗nˆ0.5/(3∗c1))): Estimated_Cpm:=c1; p_Value:=evalf(pV(c1));

The output is:

L SL:= 2.40 USL:= 3.40 T:= 2.90 C:= 1.00 n:= 100 X_bar:= 2.825 Sn:= .125 d:= .500000000 ξ := −.6000000000 Estimated_C pm:= 1.143323901 p_Value:= .03376008156

4 Parameter

ξ and the critical value c

0

Since the process parameters µ and σ are unknown, then the distribution characteristic parameter ξ = (µ − T)/σ is also un-known, which has to be estimated in real applications, naturally by substituting µ and σ by ¯X and Sn. Such approach

intro-duces additional sampling errors from estimatingξ in calculating the p-values as well as in finding the critical values, and cer-tainly would make our approach (and all existing methods) less reliable.

To eliminate the need for estimating the distribution charac-teristic parameter ξ, we examine the behavior of the p-values and the critical values c0 against the parameterξ. We find that

the p-value forξ = 0 is greater than all the p-values for other choices ofξ in all cases. Without having to estimate the unknown parameterξ we can set ξ = 0 and calculate the corresponding

p-value for an estimated index ˆCpm= c∗using the program

pro-vided in Sect. 3. Decisions made based on the conservative p-values are ensured to be more reliable than using the existing approaches. In fact, Zimmer et al. [10] noted that one can pro-vide the largest confidence interval withξ = 0 if one is unsure of the value of ξ. Hence, our result coincides with the confi-dence interval approach provided by Zimmer et al. [10]. In the example displayed in Sect. 3, if we setξ = 0, then the program gives ˆCpm= 1.1433 and the corresponding p-value as 0.0388 for

the same input parameters: LSL= 2.40, USL = 3.40, T = 2.90, C= 1.00, n = 100, ¯X = 2.825, and Sn= 0.125.

We perform extensive calculations to obtain the critical values c0 forξ = 0(0.05)3.00, n = 10(50)300, C = 1.00, 1.33,

1.50, 1.67, 2.00, andα = 0.01, 0.025, and 0.05. Noting that pa-rameter values we investigated,ξ = 0(0.05)3.00, cover a wide range of applications with process capability Cpm≥ 0. We find

that the critical value c0 (i) is decreasing inξ for ξ ≥ 0, (ii) is

decreasing in n, (iii) obtains its maximum atξ = 0 in all cases. Hence, for practical purpose we may solve Eq. 3 withξ = 0 to obtain the required critical values, without having to estimate the parameterξ. This approach ensures that the decisions made based on those critical values are more reliable than all existing methods.

Figures 1a–5a display the surface plots of c0for 0≤ ξ ≤ 3.00

and 30≤ n ≤ 300 with α = 0.05 for C = 1.00, 1.33, 1.50, 1.67, and 2.00, respectively. Figures 1b–5b plot the curves of c0

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ver-Fig. 1. a Surface plot of c0with 0≤ ξ ≤ 3 and 30 ≤

n≤ 300 for Cpm= 1.00 and α = 0.05 b Plots of c0

versusξ for Cpm= 1.00, α = 0.05, and n = 30, 50,

70, 100, 150, 200, 250, 300 (top to bottom in plot)

Fig. 2. a Surface plot of c0with 0≤ ξ ≤ 3 and 30 ≤

sus the parameterξ (0 ≤ ξ ≤ 3.00) for sample size n = 30, 50, 70, 100(50)300 from top to bottom in plots, and C= 1.00, 1.33, 1.50, 1.67, 2.00 withα = 0.05. We also investigated the behavior

of the critical values c0versus the sample size n for other

com-monly used values ofα (0.025 and 0.01), the results appear to be the same.

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5 Testing procedure and an application

The results obtained in last section show that we can set the parameterξ = (µ − T)/σ to 0 for all cases in calculating the p-values and the critical p-values. Tables 1–5 display critical p-values c0 for C= 1.00, 1.33, 1.50, 1.67, and 2.00, with sample sizes

n= 10(5)405, and α-risk = 0.01, 0.025, 0.05. For practition-ers who do not wish to run the proposed computer program to calculate the p-value, they could simply compare the estimated index ˆCpmwith the corresponding critical value c0from the

pro-vided tables. To judge if a given process meets the capability requirement, we first determine the value of C (the capabil-ity requirement) and theα-risk. Checking the appropriate table from Tables 1–5, we may obtain the critical value c0 based on

given values ofα-risk, C, and the sample size n. If the estimated value ˆCpmis greater than the critical value c0( ˆCpm> c0), then

we conclude that the process meets the capability requirement (Cpm> C). Otherwise, we do not have sufficient information to

conclude that the process meets the present capability require-ment. In this case, we would believe that Cpm≤ C.

Procedure:

STEP 1: Decide the definition of “capable” (set the value of C), and theα-risk (normally set to 0.01, 0.025, or 0.05), the chance of wrongly concluding an incapable process as capable.

STEP 2: Calculate the value of ˆCpmfrom the sample.

STEP 3: Check the appropriate table in Appendix, and find the critical value c0based on C,α-risk and n.

STEP 4: Conclude that the process is capable (Cpm> C) if ˆCpm value is greater than the critical value c0 ( ˆCpm> c0). Otherwise, we do not have enough information to con-clude that the process is capable.

As an example, we consider a process manufacturing the chip resistors. Chip resistor is an electronic passive component commonly used on electronic circuits, providing the function of reducing the voltage, current and releasing the heat. We consider the following case taken from a factory located on an indus-trial park in Taiwan making chip resistors. One of the product characteristics is justified to be under statistical control, which

is distributed as the normal distribution. The specification lim-its are set to LSL= 31.5Ω, T = 35.0Ω, and USL = 38.5Ω. The capability requirement in the company was set to C= 1.00 (ca-pable). We note that d= (USL − LSL)/2 = 3.50. To test if the process meets the capability (quality) requirement, we must de-termine whether the process meets Cpm> 1.00. The α-risk is set to be 0.05. Thus, the chance of wrongly concluding an incapable process as capable is no greater than 5%. We take a sample of size n= 100, the sample mean is calculated as ¯X = 35.58, and the sample variance is calculated as S2_n= 0.56. We further calcu-late ˆCpm= d/{3[S2_n+ ( ¯X − T)2]1/2} = 1.232. Checking Table 1 in the Appendix, we find the critical value c0= 1.133 based on C= 1.00, α-risk = 0.05 and n = 100. Because ˆCpm= 1.232 is greater than the critical value c0= 1.133, we therefore con-clude that the process meets the present capability requirement (Cpm> 1.00) and runs under the desired quality condition.

6 Conclusion

The Taguchi index Cpm has been proposed to the manufactur-ing industry to measure process performance. But all existmanufactur-ing methods for testing Cpm require further estimation of the dis-tribution parameters when calculating the p-values and critical values causing additional sampling errors, which is unreliable. In this paper, an efficient Maple computer program is provided to calculate the p-values. Extensive calculations were performed to examine the behavior of the p-values and the critical values c0 against the distribution parameter ξ = (µ − T)/σ. We find that the p-value for ξ = 0 is greater than all the p-values for other choices ofξ in all cases. Without having to estimate the unknown parameterξ we can set ξ = 0 and calculate the cor-responding p-value for an estimated index ˆCpm= c∗ using the provided Maple program. At the same time, we find the criti-cal value c0 obtains its maximum atξ = 0 in all cases. Hence, for practical purpose we may obtain the required critical values settingξ = 0, without having to estimate the parameter ξ. This approach ensures that the decisions made based on those critical values are more reliable than all existing methods. Useful crit-ical values for commonly used capability requirements are also tabulated. A simple but practical step-by-step hypothesis-testing procedure is developed for in-plant applications.

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Acknowledgement This research was supported in part by the National Science Council of the Republic of China under grant NSC 91-2213-E-167-007.

References

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3. Hsiang TC, Taguchi G (1985) A tutorial on quality control and assurance—the Taguchi methods. ASA Annual Meeting, Las Vegas, NV

4. Kotz S, Johnson NL (1993) Process capability indices. Chapman & Hall, London

5. Kotz S, Lovelace CR (1998) Process capability indices in theory and practice. Arnold, London

6. Pearn WL, Lin PC (2004) Measuring process yield based on capability index Cpm. Int J Adv Manuf Technol 24(7):503–508

7. Cheng SW (1994) Practical implementation of the process capability indices. Qual Eng 7:239–259

8. Boyles RA (1991) The Taguchi capability index. J Qual Tech 23(1): 17–26

9. Zimmer LS, Hubele NF (1997) Quantiles of the sampling distribution of Cpm. Qual Eng 10:309–329

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Appendix: Critical values for testing C

pm

Table 1. Critical values c0for C= 1.00, n = 10(5)405 and α = 0.01, 0.025,

0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 1.978 1.755 1.594 210 1.128 1.106 1.088 15 1.694 1.548 1.438 215 1.126 1.105 1.087 20 1.557 1.445 1.358 220 1.124 1.104 1.086 25 1.473 1.381 1.309 225 1.123 1.102 1.085 30 1.417 1.337 1.274 230 1.121 1.101 1.084 35 1.376 1.305 1.249 235 1.120 1.100 1.083 40 1.344 1.280 1.229 240 1.119 1.099 1.082 45 1.319 1.260 1.213 245 1.117 1.098 1.081 50 1.298 1.244 1.200 250 1.116 1.097 1.080 55 1.280 1.230 1.189 255 1.115 1.096 1.080 60 1.266 1.218 1.179 260 1.113 1.095 1.079 65 1.253 1.208 1.171 265 1.112 1.094 1.078 70 1.242 1.199 1.164 270 1.111 1.093 1.077 75 1.232 1.191 1.157 275 1.110 1.092 1.076 80 1.223 1.184 1.151 280 1.109 1.091 1.076 85 1.215 1.177 1.146 285 1.108 1.090 1.075 90 1.208 1.171 1.142 290 1.107 1.089 1.074 95 1.201 1.166 1.137 295 1.106 1.088 1.074 100 1.195 1.161 1.133 300 1.105 1.087 1.073 105 1.190 1.157 1.130 305 1.104 1.087 1.072 110 1.185 1.153 1.126 310 1.103 1.086 1.072 115 1.180 1.149 1.123 315 1.102 1.085 1.071 120 1.175 1.145 1.120 320 1.101 1.084 1.070 125 1.171 1.142 1.118 325 1.100 1.84 1.070 130 1.168 1.139 1.115 330 1.100 1.083 1.069 135 1.164 1.136 1.113 335 1.099 1.082 1.069 140 1.161 1.133 1.110 340 1.098 1.82 1.068 145 1.157 1.131 1.108 345 1.097 1.081 1.068 150 1.154 1.128 1.106 350 1.096 1.081 1.067 155 1.151 1.126 1.104 355 1.096 1.080 1.067 160 1.149 1.124 1.102 360 1.095 1.079 1.066 165 1.146 1.121 1.101 365 1.094 1.079 1.066 170 1.144 1.119 1.099 370 1.094 1.078 1.065 175 1.141 1.117 1.098 375 1.093 1.078 1.065 180 1.139 1.116 1.096 380 1.092 1.077 1.064 185 1.137 1.114 1.095 385 1.092 1.077 1.064 190 1.135 1.112 1.093 390 1.091 1.076 1.063 195 1.133 1.111 1.092 395 1.090 1.075 1.063 200 1.131 1.109 1.091 400 1.090 1.075 1.063 205 1.129 1.108 1.089 405 1.089 1.074 1.062

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Table 2. Critical values c0for C= 1.33, n = 10(5)405 and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 2.630 2.335 2.119 210 1.500 1.471 1.447 15 2.253 2.059 1.912 215 1.497 1.469 1.446 20 2.070 1.921 1.806 220 1.495 1.468 1.444 25 1.959 1.836 1.740 225 1.493 1.466 1.443 30 1.884 1.778 1.694 230 1.491 1.464 1.442 35 1.829 1.735 1.661 235 1.489 1.463 1.440 40 1.787 1.702 1.634 240 1.488 1.461 1.439 45 1.754 1.676 1.613 245 1.486 1.460 1.438 50 1.726 1.654 1.596 250 1.484 1.458 1.437 55 1.703 1.635 1.581 255 1.482 1.457 1.436 60 1.683 1.620 1.568 260 1.481 1.456 1.434 65 1.666 1.606 1.557 265 1.479 1.454 1.433 70 1.651 1.594 1.548 270 1.478 1.453 1.432 75 1.638 1.584 1.539 275 1.476 1.452 1.431 80 1.626 1.574 1.531 280 1.475 1.451 1.430 85 1.616 1.566 1.524 285 1.473 1.449 1.429 90 1.606 1.558 1.518 290 1.472 1.448 1.429 95 1.597 1.551 1.512 295 1.471 1.447 1.428 100 1.589 1.544 1.507 300 1.469 1.446 1.427 105 1.582 1.538 1.502 305 1.468 1.445 1.426 110 1.575 1.533 1.498 310 1.467 1.444 1.425 115 1.569 1.528 1.494 315 1.466 1.443 1.424 120 1.563 1.523 1.490 320 1.464 1.442 1.423 125 1.558 1.519 1.486 325 1.463 1.441 1.423 130 1.553 1.514 1.483 330 1.462 1.440 1.422 135 1.548 1.511 1.480 335 1.461 1.439 1.421 140 1.543 1.507 1.477 340 1.460 1.439 1.420 145 1.539 1.503 1.474 345 1.459 1.438 1.420 150 1.535 1.500 1.471 350 1.458 1.437 1.419 155 1.531 1.497 1.469 355 1.457 1.436 1.418 160 1.528 1.494 1.466 360 1.456 1.435 1.418 165 1.524 1.491 1.464 365 1.455 1.435 1.417 170 1.521 1.489 1.462 370 1.454 1.434 1.417 175 1.518 1.486 1.460 375 1.453 1.433 1.416 180 1.515 1.484 1.458 380 1.453 1.432 1.415 185 1.512 1.481 1.456 385 1.452 1.432 1.415 190 1.509 1.479 1.454 390 1.451 1.431 1.414 195 1.507 1.477 1.452 395 1.450 1.430 1.414 200 1.504 1.475 1.450 400 1.449 1.430 1.413 205 1.502 1.473 1.449 405 1.448 1.429 1.412

0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 2.966 2.633 2.390 210 1.691 1.659 1.632 15 2.541 2.322 2.156 215 1.689 1.657 1.631 20 2.335 2.167 2.037 220 1.686 1.655 1.629 25 2.210 2.071 1.963 225 1.684 1.653 1.627 30 2.125 2.006 1.911 230 1.682 1.651 1.626 35 2.063 1.957 1.873 235 1.680 1.650 1.624 40 2.016 1.920 1.843 240 1.678 1.648 1.623 45 1.978 1.890 1.819 245 1.676 1.646 1.622 50 1.947 1.865 1.799 250 1.674 1.645 1.620 55 1.920 1.844 1.783 255 1.672 1.643 1.619 60 1.898 1.827 1.769 260 1.670 1.642 1.618 65 1.879 1.811 1.756 265 1.668 1.640 1.617 70 1.862 1.798 1.745 270 1.666 1.639 1.615 75 1.847 1.786 1.736 275 1.665 1.637 1.614 80 1.834 1.775 1.727 280 1.663 1.636 1.613 85 1.822 1.766 1.719 285 1.662 1.635 1.612 90 1.811 1.757 1.712 290 1.660 1.633 1.611 95 1.802 1.749 1.706 295 1.659 1.632 1.610 100 1.793 1.742 1.700 300 1.657 1.631 1.609 105 1.784 1.735 1.694 305 1.656 1.630 1.608 110 1.777 1.729 1.689 310 1.654 1.629 1.607 115 1.770 1.723 1.685 315 1.653 1.628 1.606 120 1.763 1.718 1.680 320 1.652 1.626 1.605 125 1.757 1.713 1.676 325 1.650 1.625 1.604 130 1.751 1.708 1.672 330 1.649 1.624 1.604 135 1.746 1.704 1.669 335 1.648 1.623 1.603 140 1.741 1.699 1.665 340 1.647 1.622 1.602 145 1.736 1.696 1.662 345 1.646 1.621 1.601 150 1.731 1.692 1.659 350 1.644 1.621 1.600 155 1.727 1.688 1.656 355 1.643 1.620 1.600 160 1.723 1.685 1.653 360 1.642 1.619 1.599 165 1.719 1.682 1.651 365 1.641 1.618 1.598 170 1.715 1.679 1.648 370 1.640 1.617 1.597 175 1.712 1.676 1.646 375 1.639 1.616 1.597 180 1.709 1.673 1.644 380 1.638 1.615 1.596 185 1.705 1.671 1.642 385 1.637 1.615 1.595 190 1.702 1.668 1.640 390 1.636 1.614 1.595 195 1.699 1.666 1.638 395 1.635 1.613 1.594 200 1.697 1.663 1.636 400 1.634 1.612 1.594 205 1.694 1.661 1.634 405 1.633 1.611 1.593

(8)

Table 4. Critical values c0for C= 1.67, n = 10(5)405 and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 3.302 2.931 2.661 210 1.883 1.847 1.817 15 2.829 2.585 2.401 215 1.880 1.845 1.815 20 2.599 2.412 2.268 220 1.877 1.843 1.813 25 2.460 2.306 2.185 225 1.875 1.840 1.812 30 2.366 2.233 2.128 230 1.872 1.838 1.810 35 2.297 2.179 2.085 235 1.870 1.836 1.808 40 2.244 2.137 2.052 240 1.868 1.835 1.807 45 2.202 2.104 2.025 245 1.865 1.833 1.805 50 2.167 2.076 2.003 250 1.863 1.831 1.804 55 2.138 2.053 1.985 255 1.861 1.829 1.802 60 2.113 2.034 1.969 260 1.859 1.828 1.801 65 2.092 2.017 1.955 265 1.857 1.826 1.800 70 2.073 2.001 1.943 270 1.855 1.824 1.798 75 2.057 1.988 1.932 275 1.853 1.823 1.797 80 2.042 1.976 1.923 280 1.852 1.821 1.796 85 2.029 1.966 1.914 285 1.850 1.820 1.795 90 2.017 1.956 1.906 290 1.848 1.818 1.794 95 2.006 1.947 1.899 295 1.846 1.817 1.792 100 1.996 1.939 1.892 300 1.845 1.816 1.791 105 1.986 1.931 1.886 305 1.843 1.814 1.790 110 1.978 1.925 1.881 310 1.842 1.813 1.789 115 1.970 1.918 1.875 315 1.840 1.812 1.788 120 1.963 1.912 1.870 320 1.839 1.811 1.787 125 1.956 1.907 1.866 325 1.837 1.810 1.786 130 1.949 1.901 1.862 330 1.836 1.808 1.785 135 1.943 1.897 1.858 335 1.835 1.807 1.784 140 1.938 1.892 1.854 340 1.833 1.806 1.783 145 1.932 1.888 1.850 345 1.832 1.805 1.783 150 1.927 1.883 1.847 350 1.831 1.804 1.782 155 1.923 1.880 1.844 355 1.829 1.803 1.781 160 1.918 1.876 1.841 360 1.828 1.802 1.780 165 1.914 1.872 1.838 365 1.827 1.801 1.779 170 1.910 1.869 1.835 370 1.826 1.800 1.778 175 1.906 1.866 1.833 375 1.825 1.799 1.778 180 1.902 1.863 1.830 380 1.824 1.798 1.777 185 1.899 1.860 1.828 385 1.823 1.797 1.776 190 1.895 1.857 1.825 390 1.822 1.797 1.775 195 1.892 1.854 1.823 395 1.821 1.796 1.775 200 1.889 1.852 1.821 400 1.819 1.795 1.774 205 1.886 1.849 1.819 405 1.818 1.794 1.773

0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 3.955 3.510 3.187 210 2.255 2.212 2.176 15 3.388 3.096 2.875 215 2.252 2.209 2.174 20 3.113 2.889 2.716 220 2.248 2.207 2.172 25 2.946 2.761 2.617 225 2.245 2.204 2.170 30 2.833 2.674 2.548 230 2.242 2.202 2.168 35 2.751 2.609 2.497 235 2.239 2.199 2.166 40 2.687 2.560 2.457 240 2.237 2.197 2.164 45 2.637 2.520 2.425 245 2.234 2.195 2.162 50 2.595 2.487 2.399 250 2.231 2.193 2.160 55 2.560 2.459 2.377 255 2.229 2.191 2.159 60 2.531 2.435 2.358 260 2.226 2.189 2.157 65 2.505 2.415 2.341 265 2.224 2.187 2.155 70 2.483 2.397 2.327 270 2.222 2.185 2.154 75 2.463 2.381 2.314 275 2.219 2.183 2.152 80 2.445 2.367 2.302 280 2.217 2.181 2.151 85 2.429 2.354 2.292 285 2.215 2.179 2.149 90 2.415 2.342 2.283 290 2.213 2.178 2.148 95 2.402 2.332 2.274 295 2.211 2.176 2.147 100 2.390 2.322 2.266 300 2.209 2.174 2.145 105 2.379 2.313 2.259 305 2.207 2.173 2.144 110 2.369 2.305 2.252 310 2.206 2.171 2.143 115 2.359 2.297 2.246 315 2.204 2.170 2.141 120 2.350 2.290 2.240 320 2.202 2.168 2.140 125 2.342 2.283 2.235 325 2.200 2.167 2.139 130 2.335 2.277 2.229 330 2.199 2.166 2.138 135 2.327 2.271 2.225 335 2.197 2.164 2.137 140 2.321 2.266 2.220 340 2.195 2.163 2.136 145 2.314 2.261 2.216 345 2.194 2.162 2.135 150 2.308 2.256 2.212 350 2.192 2.161 2.134 155 2.302 2.251 2.208 355 2.191 2.159 2.133 160 2.297 2.247 2.204 360 2.189 2.158 2.132 165 2.292 2.242 2.201 365 2.188 2.157 2.131 170 2.287 2.238 2.198 370 2.187 2.156 2.130 175 2.282 2.234 2.195 375 2.185 2.155 2.129 180 2.278 2.231 2.192 380 2.184 2.154 2.128 185 2.274 2.227 2.189 385 2.183 2.153 2.127 190 2.270 2.224 2.186 390 2.181 2.151 2.126 195 2.266 2.221 2.183 395 2.180 2.150 2.125 200 2.262 2.218 2.181 400 2.179 2.149 2.125 205 2.258 2.215 2.178 405 2.178 2.148 2.124