Research Article | Open Access

Volume 2013 |Article ID 324940 | https://doi.org/10.1155/2013/324940

Wararit Panichkitkosolkul, "Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean", Journal of Probability and Statistics, vol. 2013, Article ID 324940, 11 pages, 2013. https://doi.org/10.1155/2013/324940

# Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean

Academic Editor: Shein-chung Chow
Received23 Jul 2013
Accepted25 Sep 2013
Published21 Nov 2013

#### Abstract

This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known population mean. One of the proposed confidence intervals is based on the normal approximation. The other proposed confidence intervals are the shortest-length confidence interval and the equal-tailed confidence interval. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. Simulation results have shown that all three proposed confidence intervals perform well in terms of coverage probability and expected length.

#### 1. Introduction

The coefficient of variation of a distribution is a dimensionless number that quantifies the degree of variability relative to the mean [1]. It is a statistical measure for comparing the dispersion of several variables obtained by different units. The population coefficient of variation is defined as a ratio of the population standard deviation to the population mean given by . The typical sample estimate of is given as where is the sample standard deviation, the square root of the unbiased estimator of population variance, and is the sample mean.

The coefficient of variation has been widely used in many areas such as science, medicine, engineering, economics, and others. For example, the coefficient of variation has also been employed by Ahn [2] to analyze the uncertainty of fault trees. Gong and Li [3] assessed the strength of ceramics by using the coefficient of variation. Faber and Korn [4] applied the coefficient of variation as a way of including a measure of variation in the mean synaptic response of the central nervous system. The coefficient of variation has also been used to assess the homogeneity of bone test samples to help determine the effect of external treatments on the properties of bones [5]. Billings et al. [6] used the coefficient of variation to study the impact of socioeconomic status on hospital use in New York City. In finance and actuarial science, the coefficient of variation can be used as a measure of relative risk and a test of the equality of the coefficients of variation for two stocks [7]. Furthermore, Pyne et al. [8] studied the variability of the competitive performance of Olympic swimmers by using the coefficient of variation.

Although the point estimator of the population coefficient of variation shown in (1) can be a useful statistical measure, its confidence interval is more useful than the point estimator. A confidence interval provides much more information about the population characteristic of interest than does a point estimate (e.g., Smithson [9], Thompson [10], and Steiger [11]). There are several approaches available for constructing the confidence interval for . McKay [12] proposed a confidence interval for based on the chi-square distribution; this confidence interval works well when [1317]. Later, Vangel [18] proposed a new confidence interval for , which is called a modified McKay’s confidence interval. His confidence interval is based on an analysis of the distribution of a class of approximate pivotal quantities for the normal coefficient of variation. In addition, modified McKay’s confidence interval is closely related to McKay’s confidence interval but it is usually more accurate and nearly exact under normality. Panichkitkosolkul [19] modified McKay’s confidence interval by replacing the sample coefficient of variation with the maximum likelihood estimator for a normal distribution. Sharma and Krishna [20] introduced the asymptotic distribution and confidence interval of the reciprocal of the coefficient of variation which does not require any assumptions about the population distribution to be made. Miller [21] discussed the approximate distribution of and proposed the approximate confidence interval for in the case of a normal distribution. The performance of many confidence intervals for obtained by McKay’s, Miller’s, and Sharma-Krishna’s methods was compared under the same simulation conditions by Ng [22].

Mahmoudvand and Hassani [23] proposed an approximately unbiased estimator for in a normal distribution and also used this estimator for constructing two approximate confidence intervals for the coefficient of variation. The confidence intervals for in normal and lognormal were proposed by Koopmans et al. [24] and Verrill [25]. Buntao and Niwitpong [26] also introduced an interval estimating the difference of the coefficient of variation for lognormal and delta-lognormal distributions. Curto and Pinto [27] constructed the confidence interval for when random variables are not independently and identically distributed. Recent work of Gulhar et al. [28] has compared several confidence intervals for estimating the population coefficient of variation based on parametric, nonparametric, and modified methods.

However, the population mean may be known in several phenomena. The confidence intervals of the aforementioned authors have not been used for estimating the population coefficient of variation for the normal distribution with a known population mean. Therefore, our main aim in this paper is to propose three confidence intervals for in a normal distribution with a known population mean.

The organization of this paper is as follows. In Section 2, the theoretical background of the proposed confidence intervals is discussed. The investigations of the performance of the proposed confidence interval through a Monte Carlo simulation study are presented in Section 3. A comparison of the confidence intervals is also illustrated by using an empirical application in Section 4. Conclusions are provided in the final section.

#### 2. Theoretical Results

In this section, the mean and variance of the estimator of the coefficient of variation in a normal distribution with a known population mean are considered. In addition, we will introduce an unbiased estimator for the coefficient of variation, obtain its variance, and finally construct three confidence intervals: normal approximation, shortest-length, and equal-tailed confidence intervals.

If the population mean is known to be , then the population coefficient of variation is given by . The sample estimate of is where . To find the expectation of (2), we have to prove the following lemma.

Lemma 1. Let be a random sample from normal distribution with known mean and variance and let . Then where .

Proof of Lemma 1. By definition, where .
Thus,
Let and . From Theorem B of Rice [29, page 197], the distribution of is central chi-square distribution with degrees of freedom. Similarly, the distribution of is central chi-square distribution with degrees of freedom; that is,
One can see that [30, page 181] where .
Similarly, where .
Equations (5) and (6) are equivalent. Thus, we obtain . Next, we will find the variance of :

By using Lemma 1, we can show that the mean and variance of are Note that as . Therefore, it follows that It means that is asymptotically unbiased and asymptotically consistent for . From (10), the unbiased estimator of is Using Lemma 1, the mean and variance of are given by Thus, Hence, is also asymptotically consistent for . Next, we examine the accuracy of from another point view. Let us first consider the following theorem.

Theorem 2. Let be a random sample from a probability density function , which has unknown parameter . If is an unbiased estimator of , it can be shown under very general conditions that the variance of must satisfy the inequality where is the Fisher information. This is known as the Cramér-Rao inequality. If , the estimator is said to be efficient.

Proof of Theorem 2. See [31, pages 377–379].

By setting in Theorem 2, it is easy to show that where is any unbiased estimator of . This means that the variance for the efficient estimator of is .

From (15), we will show that . The asymptotic expansion of the gamma function ratio is [32] Now, if in (19), we have Thus, we obtain Therefore, . This means that is asymptotically efficient (see (18)). In the following section, three confidence intervals for are proposed.

##### 2.1. Normal Approximation Confidence Interval

Using the normal approximate, we have Therefore, the confidence interval for based on (22) is where is the percentile of the standard normal distribution.

##### 2.2. Shortest-Length Confidence Interval

A pivotal quantity for is Converting the statement we can write Thus, the confidence interval for based on the pivotal quantity is where , , and the length of confidence interval for is defined as In order to find the shortest-length confidence interval for , the following problem has to be solved: where is the probability density function of central chi-square distribution with degrees of freedom. From Casella and Berger [33, pages 443-444], the shortest-length confidence interval for based on the pivotal quantity is determined by the value of and satisfying

Table 1 is constructed for the numerical solutions of these equations by using the R statistical software [3436].

 df Confidence levels 0.90 0.95 0.99 2 0.2065 12.5208 0.1015 15.1194 0.0200 20.8264 3 0.5654 13.1532 0.3449 15.5897 0.1140 20.9856 4 1.0200 14.1800 0.6918 16.5735 0.2937 21.8371 5 1.5352 15.3498 1.1092 17.7432 0.5461 22.9867 6 2.0930 16.5807 1.5776 18.9954 0.8567 24.2618 7 2.6828 17.8391 2.0851 20.2863 1.2143 25.6017 8 3.2981 19.1099 2.6235 21.5953 1.6107 26.9749 9 3.9343 20.3848 3.1874 22.9118 2.0394 28.3643 10 4.5883 21.6598 3.7729 24.2303 2.4958 29.7602 11 5.2573 22.9325 4.3768 25.5476 2.9760 31.1580 12 5.9397 24.2016 4.9967 26.8618 3.4771 32.5543 13 6.6337 25.4666 5.6308 28.1717 3.9968 33.9474 14 7.3382 26.7269 6.2776 29.4769 4.5329 35.3358 15 8.0521 27.9825 6.9357 30.7770 5.0840 36.7192 16 8.7745 29.2334 7.6042 32.0720 5.6487 38.0968 17 9.5047 30.4796 8.2820 33.3619 6.2256 39.4688 18 10.2421 31.7212 8.9685 34.6467 6.8139 40.8347 19 10.9861 32.9585 9.6629 35.9266 7.4126 42.1952 20 11.7362 34.1915 10.3647 37.2016 8.0209 43.5498 21 12.4919 35.4205 11.0733 38.4720 8.6383 44.8989 22 13.2530 36.6455 11.7882 39.7379 9.2640 46.2426 23 14.0191 37.8668 12.5092 40.9995 9.8976 47.5810 24 14.7899 39.0844 13.2357 42.2570 10.5385 48.9144 25 15.5650 40.2986 13.9675 43.5105 11.1864 50.2428 26 16.3443 41.5095 14.7043 44.7601 11.8408 51.5665 27 17.1275 42.7171 15.4458 46.0060 12.5014 52.8856 28 17.9144 43.9217 16.1917 47.2483 13.1678 54.2002 29 18.7049 45.1234 16.9419 48.4872 13.8397 55.5107 30 19.4987 46.3222 17.6961 49.7229 14.5170 56.8169 40 27.5919 58.1755 25.4233 61.9217 21.5331 69.6808 50 35.9012 69.8342 33.4085 73.8920 28.8879 82.2534 60 44.3661 81.3479 41.5794 85.6914 36.4863 94.6063 70 52.9501 92.7487 49.8923 97.3573 44.2711 106.7867 80 61.6290 104.0584 58.3183 108.9153 52.2044 118.8272 90 70.3860 115.2925 66.8374 120.3839 60.2597 130.7514 100 79.2086 126.4628 75.4347 131.7767 68.4177 142.5771 150 124.0372 181.6128 119.2737 187.9079 110.3262 200.6194 200 169.6646 235.9748 164.0642 243.1025 153.4834 257.4375 250 215.8057 289.8273 209.4667 297.6910 197.4440 313.4620 300 262.3132 343.3155 255.3057 351.8461 241.9776 368.9185
##### 2.3. Equal-Tailed Confidence Interval

The equal-tailed confidence interval for based on the pivotal quantity is where and are the and percentiles of the central chi-square distribution with degrees of freedom, respectively.

#### 3. Simulation Study

A Monte Carlo simulation was conducted using the R statistical software [3436] version 3.0.1 to investigate the estimated coverage probabilities and expected lengths of three proposed confidence intervals and to compare them to the existing confidence intervals. The estimated coverage probability and the expected length (based on replicates) are given by where denotes the number of simulation runs for which the population coefficient of variation lies within the confidence interval. The data were generated from a normal distribution with a known population mean and = 0.05, 0.10, 0.20, 0.33, 0.50, and 0.67 and sample sizes of 5, 10, 15, 25, 50, and 100. The number of simulation runs is equal to 50,000 and the nominal confidence levels are fixed at 0.90 and 0.95. Three existing confidence intervals are considered, namely, Miller’s [7], McKay’s [12], and Vangel’s [18].

Miller:

McKay:

Vangel: The upper McKay’s limit will have to be set to under the following condition [25]: and the upper Vangel’s limit will have to be set to under the following condition: As can be seen from Tables 2 and 3, the three proposed confidence intervals have estimated coverage probabilities close to the nominal confidence level in all cases. On the other hand, the Miller’s, McKay’s, and Vangel’s confidence intervals provide estimated coverage probabilities much different from the nominal confidence level, especially when the population coefficient of variation is large. In other words, the estimated coverage probabilities of existing confidence intervals tend to be too high. Additionally, the estimated coverage probabilities of existing confidence intervals increase as the values of get larger (i.e., for 95% McKay’s confidence interval, , 0.9522 for = 0.05; 0.9539 for = 0.10; 0.9856 for = 0.67). However, Figure 1 shows that the estimated coverage probabilities of the three proposed confidence intervals do not increase or decrease according to the values of .

 Coverage probabilities Expected lengths Miller McKay Vangel Approx. Shortest Equal-tailed Miller McKay Vangel Approx. Shortest Equal-tailed 5 0.05 0.8499 0.9066 0.8858 0.9016 0.9023 0.8979 0.0555 0.0607 0.0582 0.0741 0.0587 0.0675 0.10 0.8518 0.9069 0.8866 0.9024 0.9008 0.8988 0.1120 0.1237 0.1177 0.1482 0.1174 0.1349 0.20 0.8524 0.9130 0.8960 0.9036 0.8990 0.9006 0.2315 0.2689 0.2457 0.2963 0.2347 0.2696 0.33 0.8572 0.9258 0.9136 0.9038 0.8999 0.9001 0.4099 0.5872 0.4528 0.4895 0.3878 0.4453 0.50 0.8664 0.9430 0.9321 0.9036 0.8994 0.9001 0.6959 1.2123 0.9360 0.7409 0.5869 0.6741 0.67 0.8773 0.9578 0.9428 0.9031 0.9000 0.8992 1.0603 1.5764 1.6394 0.9947 0.7880 0.9050 10 0.05 0.8747 0.9031 0.8870 0.9020 0.8996 0.9006 0.0379 0.0396 0.0382 0.0431 0.0388 0.0416 0.10 0.8792 0.9052 0.8899 0.9024 0.9001 0.9014 0.0765 0.0804 0.0773 0.0864 0.0778 0.0833 0.20 0.8802 0.9135 0.9002 0.9013 0.8993 0.9001 0.1576 0.1686 0.1603 0.1726 0.1553 0.1664 0.33 0.8899 0.9304 0.9202 0.9017 0.9021 0.9015 0.2778 0.3140 0.2893 0.2853 0.2566 0.2750 0.50 0.8999 0.9527 0.9451 0.9007 0.9004 0.8995 0.4709 0.6575 0.5323 0.4329 0.3895 0.4174 0.67 0.9129 0.9694 0.9600 0.9018 0.8999 0.8992 0.7128 1.4205 1.0257 0.5801 0.5218 0.5593 15 0.05 0.8846 0.9010 0.8870 0.9000 0.8989 0.8988 0.0307 0.0316 0.0306 0.0333 0.0311 0.0326 0.10 0.8866 0.9065 0.8925 0.9011 0.9013 0.9001 0.0618 0.0638 0.0617 0.0666 0.0622 0.0652 0.20 0.8913 0.9127 0.9007 0.9006 0.8993 0.8987 0.1271 0.1328 0.1275 0.1330 0.1242 0.1301 0.33 0.9046 0.9308 0.9218 0.9012 0.9022 0.9013 0.2239 0.2418 0.2286 0.2200 0.2054 0.2151 0.50 0.9150 0.9544 0.9477 0.9000 0.9004 0.8991 0.3787 0.4522 0.4087 0.3338 0.3116 0.3264 0.67 0.9280 0.9725 0.9661 0.9010 0.9002 0.8999 0.5713 0.8900 0.7010 0.4466 0.4170 0.4367 25 0.05 0.8933 0.9056 0.8932 0.9029 0.9035 0.9021 0.0236 0.0240 0.0233 0.0247 0.0238 0.0244 0.10 0.8948 0.9054 0.8939 0.9010 0.9014 0.9004 0.0475 0.0485 0.0471 0.0495 0.0475 0.0489 0.20 0.9022 0.9146 0.9042 0.9028 0.9008 0.9021 0.0977 0.1003 0.0971 0.0988 0.0949 0.0976 0.33 0.9144 0.9318 0.9238 0.9005 0.9016 0.9004 0.1716 0.1796 0.1727 0.1630 0.1566 0.1610 0.50 0.9285 0.9548 0.9491 0.8976 0.8977 0.8978 0.2893 0.3207 0.3027 0.2471 0.2374 0.2440 0.67 0.9430 0.9768 0.9722 0.9018 0.8999 0.9008 0.4363 0.5418 0.4930 0.3310 0.3179 0.3268 50 0.05 0.8941 0.8992 0.8905 0.8993 0.8977 0.8989 0.0166 0.0167 0.0164 0.0170 0.0166 0.0168 0.10 0.8996 0.9043 0.8949 0.9004 0.9007 0.8997 0.0334 0.0337 0.0330 0.0339 0.0333 0.0337 0.20 0.9061 0.9118 0.9041 0.8994 0.8989 0.8996 0.0688 0.0697 0.0680 0.0678 0.0665 0.0674 0.33 0.9220 0.9314 0.9253 0.8997 0.8996 0.8994 0.1206 0.1236 0.1204 0.1118 0.1096 0.1112 0.50 0.9436 0.9583 0.9539 0.9009 0.9022 0.9010 0.2031 0.2153 0.2084 0.1695 0.1662 0.1685 0.67 0.9588 0.9801 0.9770 0.9010 0.9009 0.9008 0.3062 0.3460 0.3309 0.2271 0.2226 0.2257 100 0.05 0.8998 0.9026 0.8961 0.9019 0.8902 0.9017 0.0117 0.0117 0.0115 0.0118 0.0113 0.0118 0.10 0.9000 0.9031 0.8959 0.8995 0.8878 0.8992 0.0236 0.0237 0.0233 0.0236 0.0227 0.0236 0.20 0.9110 0.9131 0.9072 0.9011 0.8901 0.9008 0.0485 0.0488 0.0480 0.0472 0.0453 0.0471 0.33 0.9277 0.9329 0.9282 0.9015 0.8900 0.9012 0.0850 0.0863 0.0847 0.0779 0.0748 0.0777 0.50 0.9485 0.9589 0.9561 0.9001 0.8885 0.8998 0.1429 0.1486 0.1455 0.1180 0.1133 0.1177 0.67 0.9678 0.9810 0.9790 0.9020 0.8910 0.9021 0.2157 0.2347 0.2289 0.1582 0.1519 0.1578
 Coverage probabilities Expected lengths Miller McKay Vangel Approx. Shortest Equal-tailed Miller McKay Vangel Approx. Shortest Equal-tailed 5 0.05 0.8829 0.9533 0.9440 0.9538 0.9504 0.9511 0.0661 0.0785 0.0762 0.1058 0.0758 0.0870 0.10 0.8827 0.9537 0.9457 0.9549 0.9501 0.9506 0.1333 0.1608 0.1544 0.2113 0.1513 0.1737 0.20 0.8847 0.9578 0.9508 0.9548 0.9500 0.9507 0.2756 0.3630 0.3282 0.4226 0.3026 0.3475 0.33 0.8904 0.9647 0.9599 0.9542 0.9501 0.9501 0.4880 0.8954 0.6498 0.6986 0.5001 0.5743 0.50 0.8934 0.9711 0.9656 0.9537 0.9487 0.9491 0.8276 1.3796 1.4333 1.0561 0.7561 0.8683 0.67 0.9042 0.9795 0.9721 0.9548 0.9495 0.9502 1.2550 1.5758 1.9791 1.4140 1.0124 1.1625 10 0.05 0.9115 0.9522 0.9440 0.9511 0.9502 0.9495 0.0451 0.0490 0.0478 0.0551 0.0480 0.0515 0.10 0.9125 0.9539 0.9460 0.9529 0.9510 0.9505 0.0912 0.0997 0.0968 0.1105 0.0962 0.1031 0.20 0.9156 0.9588 0.9522 0.9521 0.9506 0.9498 0.1881 0.2113 0.2023 0.2209 0.1924 0.2062 0.33 0.9201 0.9663 0.9620 0.9507 0.9499 0.9489 0.3311 0.4052 0.3718 0.3645 0.3174 0.3401 0.50 0.9281 0.9788 0.9751 0.9510 0.9492 0.9500 0.5606 0.9797 0.7415 0.5528 0.4814 0.5159 0.67 0.9372 0.9856 0.9812 0.9504 0.9500 0.9492 0.8470 1.7544 1.5776 0.7398 0.6442 0.6904 15 0.05 0.9244 0.9517 0.9443 0.9507 0.9506 0.9499 0.0366 0.0386 0.0377 0.0415 0.0380 0.0398 0.10 0.9250 0.9523 0.9446 0.9494 0.9501 0.9475 0.0737 0.0781 0.0761 0.0830 0.0760 0.0796 0.20 0.9294 0.9592 0.9537 0.9516 0.9509 0.9507 0.1520 0.1637 0.1584 0.1660 0.1521 0.1593 0.33 0.9324 0.9681 0.9634 0.9502 0.9493 0.9489 0.2669 0.3016 0.2861 0.2737 0.2508 0.2626 0.50 0.9418 0.9811 0.9783 0.9501 0.9490 0.9497 0.4495 0.5917 0.5267 0.4141 0.3794 0.3973 0.67 0.9528 0.9894 0.9862 0.9496 0.9510 0.9493 0.6819 1.3361 1.0306 0.5564 0.5097 0.5338 25 0.05 0.9356 0.9513 0.9458 0.9504 0.9500 0.9505 0.0281 0.0290 0.0284 0.0302 0.0287 0.0295 0.10 0.9338 0.9509 0.9452 0.9491 0.9481 0.9484 0.0566 0.0586 0.0573 0.0604 0.0574 0.0590 0.20 0.9383 0.9580 0.9527 0.9497 0.9495 0.9491 0.1167 0.1219 0.1188 0.1209 0.1149 0.1181 0.33 0.9453 0.9701 0.9664 0.9510 0.9497 0.9505 0.2043 0.2192 0.2118 0.1990 0.1892 0.1945 0.50 0.9575 0.9839 0.9816 0.9520 0.9516 0.9512 0.3456 0.4004 0.3783 0.3024 0.2875 0.2956 0.67 0.9651 0.9920 0.9899 0.9512 0.9503 0.9505 0.5205 0.7126 0.6382 0.4045 0.3845 0.3953 50 0.05 0.9400 0.9504 0.9458 0.9504 0.9488 0.9499 0.0198 0.0201 0.0197 0.0204 0.0199 0.0202 0.10 0.9431 0.9520 0.9473 0.9493 0.9491 0.9492 0.0398 0.0405 0.0398 0.0409 0.0399 0.0405 0.20 0.9479 0.9581 0.9534 0.9496 0.9491 0.9491 0.0819 0.0837 0.0821 0.0817 0.0797 0.0808 0.33 0.9581 0.9695 0.9669 0.9506 0.9510 0.9502 0.1437 0.1490 0.1457 0.1349 0.1316 0.1334 0.50 0.9686 0.9853 0.9834 0.9518 0.9512 0.9514 0.2420 0.2615 0.2538 0.2044 0.1994 0.2022 0.67 0.9776 0.9940 0.9927 0.9507 0.9506 0.9510 0.3652 0.4272 0.4089 0.2740 0.2673 0.2710 100 0.05 0.9454 0.9502 0.9463 0.9496 0.9496 0.9492 0.0139 0.0140 0.0138 0.0141 0.0140 0.0141 0.10 0.9479 0.9528 0.9494 0.9511 0.9502 0.9507 0.0281 0.0283 0.0279 0.0283 0.0280 0.0282 0.20 0.9545 0.9590 0.9554 0.9500 0.9501 0.9501 0.0578 0.0584 0.0576 0.0566 0.0559 0.0563 0.33 0.9621 0.9697 0.9675 0.9493 0.9489 0.9491 0.1013 0.1034 0.1018 0.0934 0.0923 0.0929 0.50 0.9758 0.9844 0.9834 0.9489 0.9486 0.9488 0.1705 0.1789 0.1757 0.1416 0.1399 0.1408 0.67 0.9849 0.9946 0.9939 0.9495 0.9489 0.9492 0.2570 0.2840 0.2775 0.1896 0.1873 0.1886

As can be seen from Figure 2, McKay’s and Vangel’s confidence intervals have longer expected lengths than Miller’s and the proposed confidence intervals. While the expected lengths of the three proposed confidence intervals are shorter than the lengths of the existing ones in almost all cases. Additionally, when the sample sizes increase, the lengths become shorter (i.e., for 95% shortest-length confidence interval, = 0.20, 0.1553 for ; 0.0949 for = 25; 0.0665 for = 50).

#### 4. An Empirical Application

To illustrate the application of the confidence intervals proposed in the previous section, we used the weights (in grams) of 61 one-month old infants listed as follows:The data are taken from the study by Ziegler et al. [37] (cited in Ledolter and Hogg [38], page 287). The histogram, density plot, Box-and-Whisker plot, and normal quantile-quantile plot are displayed in Figure 3. Algorithm 1 shows the result of the Shapiro-Wilk normality test.

 Shapiro-Wilk normality test data: weight W = 0.978, P-value = 0.3383

As they appear in Figure 3 and Algorithm 1, we find that the data are in excellent agreement with a normal distribution. From past research, we assume that the population mean of the weight of one-month old infants is about 4400 grams. An unbiased estimator of the coefficient of variation is . The 95% of proposed and existing confidence intervals for the coefficient of variation are calculated and reported in Table 4. This result confirms that the three confidence intervals proposed in this paper are more efficient than the existing confidence intervals in terms of length of interval.

 Methods Confidence intervals Lengths Lower limit Upper limit Miller 0.1131 0.1635 0.0504 McKay 0.1163 0.1675 0.0512 Vangel 0.1162 0.1674 0.0511 Normal approx. 0.1179 0.1689 0.0510 Shortest 0.1159 0.1659 0.0500 Equal-tailed 0.1175 0.1681 0.0506

#### 5. Conclusions

The coefficient of variation is the ratio of standard deviation to the mean and provides a widely used unit-free measure of dispersion. It can be useful for comparing the variability between groups of observations. Three confidence intervals for the coefficient of variation in a normal distribution with a known population mean have been developed. The proposed confidence intervals are compared with Miller’s, McKay’s, and Vangel’s confidence intervals through a Monte Carlo simulation study. Normal approximation, shortest-length, and equal-tailed confidence intervals are better than the existing confidence intervals in terms of the expected length and the closeness of the estimated coverage probability to the nominal confidence level.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author is grateful to Professor Dr. Tonghui Wang, Professor Dr. John J. Borkowski, and anonymous referees for their valuable comments and suggestions, which have significantly enhanced the quality and presentation of this paper.

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