Confidence Intervals for Mean and Difference of Means of Normal Distributions with Unknown Coefficients of Variation

This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.


Introduction
It is well known that the sample mean, x, is the uniformly minimum variance unbiased (UMVU) estimator of the normal population mean µ; see the paper by Sahai et al. [1].Dropping the requirement of unbiasedness, Searls [2] proposed the minimum mean squared error (MMSE) estimator for normal mean with known coefficient of variation (CV).Khan [3] discussed the estimation of the mean with known CV in one sample case.Gleser and Healy [4] proposed the minimum quadratic risk scale-invariant estimator for the normal mean with known CV.Bhat and Rao [5] investigated the tests for a normal mean with known CV.Niwitpong et al. [6] provided confidence intervals for the difference between normal population means with known CVs.Niwitpong [7] presented confidence intervals for the normal mean with known CV.Niwitpong and Niwitpong [8] proposed the confidence interval for the normal mean with a known CV based on the best unbiased estimator, which was proposed by Khan [3].Niwitpong [9] proposed the confidence interval for the normal mean with a known CV based on the t-test.Niwitpong and Niwitpong [10] constructed new confidence intervals for the difference between normal means with known CV.Sodanin et al. [11] proposed confidence intervals for the common mean of normal distributions with known CV.
In practice, the CV is unknown.Furthermore, the CV needs to be estimated.Therefore, Srivastava [12] proposed a UMVU estimator for the estimation of the normal mean with unknown CV, θ = x/(1 + (s 2 /(n x2 ))), where CV is defined as σ/µ.The UMVU estimator, estimated from the MMSE estimator of Searls [2], is more efficient than the usual unbiased estimator sample mean x whenever σ 2 /(µσ 2 ) is at least 0.5.Srivastava and Singh [13] provided a UMVU estimate of the relative efficiency ratio of θ.Moreover, Sahai [14] developed a new estimator for the normal mean with unknown CV.Sahai and Acharya [15] studied the iterative estimation of the normal population mean using computational-statistical intelligence.However, a confidence interval provides more information about a population value of the quantity than a point estimate.Therefore, it is of practical and theoretical importance to develop procedures for confidence interval estimation of the mean of the normal distribution with unknown CV.Hence, along similar lines as Srivastava [12], we construct the new confidence intervals for the normal mean with unknown CV and compare with the standard confidence intervals: the Student's t-distribution and the z-distribution.The comparison can be based on coverage probability, as well as the length of the confidence intervals.The average length of the confidence intervals could also be analytically obtained and hence compared; see, e.g., Sodanin et al. [16], who proposed the confidence intervals for the normal population mean with unknown CV based on the generalized confidence interval (GCI) approach.This paper extends the work of Sodanin et al. [16] to construct confidence intervals for the normal population mean with unknown CV based on the GCI approach and the new confidence intervals based on the large sample (LS) approach.Furthermore, three new confidence intervals for the difference between normal means with unknown CVs were also proposed based on the GCI approach, the LS approach and the method of variance estimates recovery (MOVER) approach and compared with the well-known Welch-Satterthwaite (WS) approach.For more on confidence intervals on CV, we refer our readers to Banik and Kibria [17], Gulhar et al. [18] and, recently, Albatineh et al. [19], among others.
This paper is organized as follows.In Section 2, the confidence intervals for the single normal mean with unknown CV are presented.In Section 3, the confidence intervals for the difference between normal means with unknown CVs are provided.In Section 4, simulation results are presented to evaluate the coverage probabilities and average lengths in the comparison of the proposed approaches.In Section 5, the proposed approaches are illustrated using three examples.Section 6 summarizes this paper.

Confidence Intervals for the Mean of the Normal Distribution with Unknown Coefficient of Variation
Suppose that X = (X 1 , X 2 , . . ., X n ) are independent random variables each having the normal distribution with mean µ and variance σ 2 .The CV is defined by τ = σ/µ.Let X and S 2 be the sample mean and sample variance for X, respectively.Furthermore, let x and s 2 be the observed sample of X and S 2 , respectively.Searls [2] proposed the MMSE estimator for the normal population mean with variance, θ, defined by: However, the CV needs to be estimated.Srivastava [12] proposed an estimator of the mean with unknown CV, which is defined by: Moreover, Sahai [14] proposed an alternative estimator of the normal population mean with unknown CV, which is defined by: The estimator of θ * is defined by: Theorem 1. Suppose that X = (X 1 , X 2 , . . ., X n ) is a random sample from N(µ, σ 2 ).Suppose X and S 2 are a sample mean and a sample variance, respectively.Let θ * be an estimator of the normal population mean with unknown CV, and let θ * be an estimator of θ * .The mean and variance of θ * are obtained by: and: According to Thangjai et al. [20], the mean of X2 is computed by the moment generating function, and the variance of X2 is computed by Stein's lemma.Therefore, the mean and the variance of X2 are defined by: From Thangjai et al. [20], the mean and the variance of S 2 / X2 are defined by: 2 and: Therefore, the mean and variance of n − (S 2 / X2 ) are defined by: 2 and: According to Blumenfeld [21], the mean and variance of θ * are obtained by: ) be an estimator of the mean with unknown CV, and let θ * = n X/(n − (S 2 / X2 )) be an estimator of θ * .From Theorem 1, θ * is distributed normally with mean τ * 1 and variance τ * 2 , which is defined by: where: and: Proof.For the proof of the mean and variance of θ is similarly to Theorem 1. Proposition 2. Let X = (X 1 , X 2 , . . ., X n ) be a random sample from the normal distribution with the mean µ and the variance σ 2 .Let X and S 2 be the corresponding point estimates of µ and σ 2 .Then: where θ = n X/(n + (S 2 / X2 )), τ 1 is E θ in Equation ( 8), and τ 2 is Var θ in Equation (9).
Proof.For the proof of the distribution of θ is similar to Proposition 1.

Generalized Confidence Intervals for the Mean of the Normal Distribution with Unknown Coefficient of Variation
Definition 1.Let X = (X 1 , X 2 , . . . ,X n ) be a random sample from a distribution F (x|δ), which depends on a vector of parameters δ = (θ, ϑ) where θ is the parameter of interest and ϑ is possibly a vector of nuisance parameters.Weerahandi [22] defines a generalized pivot R (X, x, θ, ϑ) for confidence interval estimation, where x is an observed value of X, as a random variable having the following two properties: (i) R (X, x, θ, ϑ) has a probability distribution that is free of unknown parameters.(ii) The observed value of R (X, x, θ, ϑ), X = x, is the parameter of interest.

Recall that:
(n − 1) S 2 where V is chi-squared distribution with n − 1 degrees of freedom.Now, write: The generalized pivotal quantity (GPQ) for σ 2 is defined by: Moreover, the mean is given by: where Z and U denote the standard normal distribution and chi-square distribution with n − 1 degrees of freedom, respectively.Thus, the GPQ for µ is defined by: Therefore, the GPQ for θ is defined by: Moreover, the GPQ for θ * is defined by: Therefore, the 100 (1 − α) % two-sided confidence intervals for the single normal mean with unknown CV based on the GCI approach are obtained by: and: where R θ (α) and R θ * (α) denote the 100 (α)-th percentiles of R θ and R θ * , respectively.
Algorithm 1.For a given x, the GCI for θ and θ * can be computed by the following steps: Step 1. Generate V ∼ χ 2 n−1 , and then, compute R σ 2 from Equation (13).
Step 4. Repeat Steps 1-3 a total q times, and obtain an array of R θ 's and R θ * 's. Step

Large Sample Confidence Intervals for the Mean of the Normal Distribution with Unknown Coefficient of Variation
Again, from Equations ( 2) and ( 4), the estimators of the mean with unknown CV are defined by: and: From Theorem 2, the variance of θ is defined by: , (22) with µ and σ 2 replaced by x and s 2 , respectively.
From Theorem 1, the variance of θ * is defined by: with µ and σ 2 replaced by x and s 2 , respectively.
Therefore, the 100 (1 − α) % two-sided confidence intervals for the single normal mean with unknown CV based on the LS approach are obtained by: and: where z 1−α/2 denotes the 1 − α/2-th quantile of the standard normal distribution.
Algorithm 2. The coverage probability for θ and θ * can be computed by the following steps: Step 1. Generate X 1 , X 2 , . . ., X n from N µ, σ 2 and then compute x and s 2 .
Step 2. Use Algorithm 1 to construct CI GCI.θ and record whether or not the value of θ falls in the corresponding confidence interval.
Step 3. Use Algorithm 1 to construct CI GCI.θ * and record whether or not the value of θ * falls in the corresponding confidence interval.
Step 4. Use Equation (24) to construct CI LS.θ and record whether or not the value of θ falls in the corresponding confidence interval.
Step 5. Use Equation (25) to construct CI LS.θ * and record whether or not the value of θ * falls in the corresponding confidence interval.
Step 6. Repeat Steps 1-5, a total M times.Then, for CI GCI.θ and CI LS.θ , the fraction of times that all θ are in their corresponding confidence intervals provides an estimate of the coverage probability.Similarly, for CI GCI.θ * and CI LS.θ * , the fraction of times that all θ * are in their corresponding confidence intervals provides an estimate of the coverage probability.

Confidence Intervals for the Difference between the Means of Normal Distributions with Unknown Coefficients of Variation
Suppose that X = (X 1 , X 2 , . . . ,X n ) are independent random variables each having a normal distribution with mean µ X and variance σ 2 X .Additionally, suppose that Y = (Y 1 , Y 2 , . . . ,Y m ) are independent random variables each having a normal distribution with mean µ Y and variance σ 2 Y .Furthermore, X and Y are independent.Let X and S 2 X be the sample mean and the sample variance for X, respectively.Furthermore, let x and s 2 X be the observed sample of X and S 2 X , respectively.Similarly, let Ȳ and S 2 Y be the sample mean and the sample variance for Y, respectively.Furthermore, let ȳ and s 2 Y be the observed sample of Ȳ and S 2 Y , respectively.Let δ = θ X − θ Y be the difference between means with unknown CVs.The estimators of δ are defined by: and: where θX and θ * X denote the estimator of θ X and θ * X , respectively, and θY and θ * Y denote the estimator of θ Y and θ * Y , respectively.
Let X and Y be independent.Let X and S 2 X be the sample mean and the sample variance for X, respectively.Furthermore, let Ȳ and S 2 Y be the sample mean and the sample variance for Y, respectively.Let θ X and θ Y be the mean with unknown CV of X and Y, respectively.Let δ be the difference between θ X and θ Y .Let δ be an estimator of δ.The mean and variance of δ are obtained by: and: Proof.Let δ = θ X − θ Y be the difference between means with unknown CVs.Let δ be an estimator of δ, which is defined by: δ Thus, the mean and variance of δ are obtained by: and: Hence, Theorem 3 is proven.
Let X and Y be independent.Let X and S 2 X be the sample mean and the sample variance for X, respectively.Furthermore, let Ȳ and S 2 Y be the sample mean and the sample variance for Y, respectively.Let θ * X and θ * Y be the mean with unknown CV of X and Y, respectively.Let δ * be the difference between θ * X and θ * Y .Additionally, let δ * be an estimator of δ * .The mean and variance of δ * are obtained by: and: Proof.For the proof of the mean and variance of δ * is similar to Theorem 3.

Generalized Confidence Intervals for the Difference between Means of Normal Distributions with Unknown Coefficients of Variation
From the random variable X and Y, since: The GPQs for σ 2 X and σ 2 Y are defined by: Moreover, the means are given by: Thus, the GPQs for µ X and µ Y are defined by: Therefore, the GPQ for δ is defined by: Moreover, the GPQ for δ * is defined by: Therefore, the 100 (1 − α) % two-sided confidence intervals for the difference between normal means with unknown CVs based on the GCI approach are obtained by: and: where R δ (α) and R δ * (α) denote the 100 (α)-th percentiles of R δ and R δ * , respectively.
Algorithm 3.For a given x and ȳ, the GCI for δ and δ * can be computed by the following steps: Step Step 3. Compute R δ from Equation (36), and compute R δ * from Equation (37).
Step 4. Repeat Steps 1-3, a total q times, and obtain an array of R δ 's and R δ * 's. Step

Large Sample Confidence Intervals for the Difference between Means of Normal Distributions with Unknown Coefficients of Variation
Again, the estimators of the difference between means with unknown CVs are defined by: From Theorem 3, the variance of δ is defined by: with µ X , µ Y , σ 2 X and σ 2 Y replaced by x, ȳ, s 2 X and s 2 Y , respectively.From Theorem 4, the variance of δ * is defined by: with µ X , µ Y , σ 2 X and σ 2 Y replaced by x, ȳ, s 2 X and s 2 Y , respectively.Therefore, the 100 (1 − α) % two-sided confidence intervals for the difference between normal means with unknown CVs based on the LS approach are obtained by: and: where z 1−α/2 denotes the 1 − α/2-th quantile of the standard normal distribution.

Method of Variance Estimates Recovery Confidence Intervals for the Difference between Means of Normal Distributions with Unknown Coefficients of Variation
Since the difference between means is denoted by δ = θ X − θ Y , where θ X and θ Y are the means of X = (X 1 , X 2 , . . . ,X n ) and Y = (Y 1 , Y 2 , . . . ,Y m ), respectively, suppose that θX and θY are estimators of θ X and θ Y , respectively.The confidence intervals for θ X and θ Y are defined by: and: Similarly, the difference between means is denoted by δ The confidence intervals for θ * X and θ * Y are defined by: and: The MOVER approach, introduced by Donner and Zou [23], is used to construct the 100 (1 − α) % two-sided confidence interval (L δ , U δ ) of θ X − θ Y where L δ and U δ denote the lower limit and upper limit of the confidence interval, respectively.The lower limit and upper limit for δ are given by: and: (51) Similarly, the lower limit and upper limit for δ * are given by: and: Therefore, the 100 (1 − α) % two-sided confidence intervals for the difference between normal means with unknown CVs based on the MOVER approach are obtained by: and: Algorithm 4. The coverage probability for δ and δ * can be computed by the following steps: Step 1. Generate X 1 , X 2 , . . ., X n from N µ X , σ 2 X , and then, compute x and s 2 X .Additionally, generate Y , and then, compute ȳ and s 2 Y .
Step 2. Use Algorithm 3 to construct CI GCI.δ , and record whether or not the values of δ fall in the corresponding confidence interval.
Step 3. Use Algorithm 3 to construct CI GCI.δ * , and record whether or not the values of δ * fall in the corresponding confidence interval.
Step 4. Use Equation (44) to construct CI LS.δ , and record whether or not the values of δ fall in the corresponding confidence interval.
Step 5. Use Equation (45) to construct CI LS.δ * , and record whether or not the values of δ * fall in the corresponding confidence interval.
Step 6. Use Equation (54) to construct CI MOVER.δ , and record whether or not the values of δ fall in the corresponding confidence interval.
Step 7. Use Equation (55) to construct CI MOVER.δ* , and record whether or not the values of δ * fall in the corresponding confidence interval.
Step 8. Repeat Steps 1-7, a total M times.Then, for CI GCI.δ , CI LS.δ and CI MOVER.δ , the fraction of times that all δ are in their corresponding confidence intervals provides an estimate of the coverage probability.Similarly, for CI GCI.δ * , CI LS.δ * and CI MOVER.δ* , the fraction of times that all δ * are in their corresponding confidence intervals provides an estimate of the coverage probability.

Simulation Studies
To compare the performance of the confidence intervals, coverage probabilities and average lengths, introduced in Sections 2 and 3, two simulation studies were conducted.Comparison studies were also conducted using the Student's t-distribution, the z-distribution and the WS approach.The Student's t-distribution was used to construct the confidence interval for the single mean of the normal distribution when the sample size is small, whereas the z-distribution was used to construct the confidence interval when the sample size is large.The WS approach was used for constructing the confidence interval for the difference of the means of the normal distribution; see the paper by Niwitpong and Niwitpong [24].The nominal confidence level of 1 − α = 0.95 was set.The confidence interval, with the values of the coverage probability greater than or close to the nominal confidence level and also having the shortest average length, was chosen.
Firstly, the performances of the confidence intervals for the single mean of the normal distribution with unknown CV (θ and θ * ) were compared.The confidence intervals were constructed with the GCI approach (CI GCI.θ and CI GCI.θ * ) and the LS approach (CI LS.θ and CI LS.θ * ).Furthermore, the standard confidence interval for the single mean of the normal distribution (CI µ ) was constructed based on the Student's t-distribution and the z-distribution.Algorithm 1 and Algorithm 2 were used to compute coverage probabilities and average lengths with q = 2500 and M = 5000 of sample size n from N(µ, σ 2 ) for µ = 1.0, σ = 0.3, 0.5, 0.7, 0.9, 1.0, 1.1, 1.3, 1.5, 1.7, 2.0 and n = 10, 20, 30, 50, 100.The CVs were computed by σ/µ.Tables 1 and 2 show the coverage probabilities and average lengths of the 95% two-sided confidence intervals for θ, θ * and µ.The results indicated that the GCIs are similar to the paper by Sodanin et al. [16] in terms of coverage probability and average length.For the GCI approach, CI GCI.θ provides better confidence interval estimates than CI GCI.θ * in almost all cases.This is because the coverage probabilities of CI GCI.θ * are close to 1.00 when σ increases.Hence, CI GCI.θ * is a conservative confidence interval when σ increases.For the LS approach, the coverage probabilities of CI LS.θ and CI LS.θ * provide less than the nominal confidence level of 0.95 and are close to 1.00 when σ increases.Therefore, the LS approach is not recommended to construct the confidence interval for the single mean of the normal distribution with unknown CV.This is then compared with CI µ .For a small sample size, the coverage probability of CI GCI.θ performs as well as that of CI µ .The length of CI µ is a bit shorter than the length of CI GCI.θ .Hence, CI µ is better than CI GCI.θ in terms of the average length when the sample size is small.For a large sample size, CI GCI.θ is better than CI µ in terms of coverage probability.Furthermore, the coverage probability of CI GCI.θ is more stable than that of CI µ in all sample size cases.The second simulation study was to compare the performance of confidence intervals for the difference between two means of normal distributions with unknown CVs (δ and δ * ).There are three approaches; GCIs are defined as CI GCI.δ and CI GCI.δ * ; large sample confidence intervals are defined as CI LS.δ and CI LS.δ * ; and MOVER confidence intervals are defined as CI MOVER.δ and CI MOVER.δ* compared with the WS confidence interval for the difference of the means of the normal distribution (CI µ X −µ Y ).Algorithm 3 and Algorithm 4 were used to compute coverage probabilities and average lengths with q = 2500 and M = 5000.The sample sizes n from N(µ X , σ 2 X ) and m from N(µ Y , σ 2 Y ) for the sample sizes were (n, m) = (10,10), (10,20), (30,30), (20,30), (50,50), (30,50), (100,100) and (50,100).The population means were µ X = µ Y = µ = 1.0, and the population standard deviations were σ X = 0.3, 0.5, 0.7, 0.9, 1.0, 1.1, 1.3, 1.5, 1.7, 2.0 and σ Y = 1.0.The coefficients of variation were computed by τ X = σ X /µ X and τ Y = σ Y /µ Y ; also, the ratio of τ X to τ Y reduces to σ X /σ Y when we set µ X = µ Y .Tables 3 and 4 show that the coverage probabilities and average lengths of 95% two-sided confidence intervals for δ, δ * and µ X − µ Y .For the GCI approach, the coverage probabilities of CI GCI.δ are close to the nominal confidence level of 0.95 for all cases.For small sample sizes, CI GCI.δ * is the conservative confidence interval because the coverage probabilities are in the range from 0.97-1.00.Moreover, the coverage probabilities of CI GCI.δ * are close to the nominal confidence level of 0.95 when the sample sizes (n and m) increase.For the LS approach, CI LS.δ have the coverage probabilities under the nominal confidence level of 0.95 and close to the nominal confidence level of 0.95 when the sample sizes are large.Furthermore, CI LS.δ * is a conservative confidence interval because the coverage probabilities are close to 1.00.For the MOVER approach, the coverage probability of CI MOVER.δ is not stable, whereas CI MOVER.δ* is a conservative confidence interval.In addition, CI GCI.δ is better than CI µ X −µ Y in terms of coverage probability.The simulation results are presented in Table 5.The coverage probability of CI GCI.θ is as good as the coverage probability of CI µ .The length of CI µ provides a bit shorter length of CI GCI.θ .Hence, the confidence interval based on the Student's t-distribution is better than the other confidence intervals when the sample size is small.Therefore, these results confirm the simulation results for a small sample size in the previous section.The simulation results are presented in Table 6.The confidence interval based on the z-distribution yields an interval length shorter than the other confidence intervals.However, the coverage probabilities of the GCI are much closer to the nominal confidence level of 0.95 than those of other confidence intervals.Therefore, the GCI approach provides the best confidence interval when the sample size is large.Hence, the results support the simulation results for large sample size in the previous section.Example 3. The data example is taken from Lee and Lin [25] and was originally given by Jarvis et al. [26] and Pagano and Gauvreau [27].The data are fitted by the normal distribution, representing carboxyhemoglobin levels for nonsmokers and cigarette smokers.The summary statistics of nonsmokers were n = 121, x = 1.3000 and s 2 X = 1.7040.For cigarette smokers, the summary statistics were m = 75, ȳ = 4.1000, and s 2 Y = 4.0540.The CVs of nonsmoker and cigarette smoker were 1.0041 and 0.4911, respectively.The difference between x and ȳ was −2.8000.The GCIs for the difference between two means with unknown CVs δ and δ * were, respectively, (−3.3269, −2.2880) and (−3.3260, −2.2956) with interval lengths of 1.0389 and 1.0304.The large sample confidence intervals for the difference between two means with unknown CVs δ and δ * were, respectively, (−5.2288, −0.3664) and (−5.8213, 0.2167) with interval lengths of 4.8624 and 6.0380.The MOVER confidence intervals for the difference between two means with unknown CVs δ and δ * were, respectively, (−5.2690, −0.3262) and (−5.8714, 0.2668) with interval lengths of 4.9428 and 6.1382.Finally, the WS confidence interval for the difference between two means µ X − µ Y was (−3.3172, −2.2828) with an interval length of 1.0344.
Table 7 presents the simulation results.The GCI approach and the WS confidence interval have yielded a minimum coverage probability at 0.95.The length of one of the GCI approach, CI GCI.δ * , is a bit shorter than the length of CI µ X −µ Y .The coverage probability of CI GCI.δ is better than that of CI µ X −µ Y .Hence, the GCI approach performs well in terms of the coverage probability.Therefore, these results confirm the simulation results in the previous section.

Discussion and Conclusions
Sodanin et al. [16] constructed the GCIs for the mean of the normal distribution with unknown CV.This paper provides generalized confidence intervals (CI GCI.θ and CI GCI.θ * ) and proposes large sample confidence intervals (CI LS.θ and CI LS.θ * ) for the single mean of the normal distribution with unknown CV (θ and θ * ).Comparison studies were also conducted using the standard confidence interval for the normal mean (CI µ ) based on the Student's t-distribution and the z-distribution, which are much more simple and easier to implement.Moreover, the new confidence intervals were proposed for the difference between two means of the normal distributions with unknown CVs (δ and δ * ).The confidence intervals for δ and δ * were constructed based on the GCI approach (CI GCI.δ and CI GCI.δ * ), the LS approach (CI LS.δ and CI LS.δ * ) and the MOVER approach (CI MOVER.δ and CI MOVER.δ* ), compared with the standard confidence interval, using the WS approach to construct the confidence interval for the difference of two means of the normal distribution (CI µ X −µ Y ).The coverage probabilities and average lengths of the proposed confidence intervals were evaluated through Monte Carlo simulations.
For the single mean with unknown CV, the results are similar to the paper by Sodanin et al. [16] in terms of coverage probability and average length for all cases.The coverage probabilities of CI GCI.θ were satisfactorily stable around 0.95.Therefore, CI GCI.θ was preferred for the single mean of the normal distribution with unknown CV.CI LS.θ and CI LS.θ * have the coverage probabilities under the nominal confidence level of 0.95 and close to 1.00 when σ increases.Therefore, the LS approach is not recommended to construct the confidence interval for the mean with unknown CV.Furthermore, CI µ is better than CI GCI.θ in terms of the average length when the sample size is small, whereas CI GCI.θ is better than CI µ in terms of coverage probability when the sample size is large.However, the coverage probability of CI GCI.θ is more stable than that of CI µ in all sample size cases.Therefore, the GCI approach is recommended as an interval estimator for the mean with unknown CV.
For the difference of two means with unknown CVs, the coverage probabilities of CI GCI.δ satisfy the nominal confidence level of 0.95 for all cases.Therefore, CI GCI.δ was preferred for the difference of the means with unknown CVs.The LS and MOVER approaches are not recommended to construct the confidence interval for the difference of means with unknown CVs.Furthermore, CI GCI.δ is better than CI µ X −µ Y in terms of the coverage probability.Therefore, the GCI approach can be used to estimate the confidence interval for the difference of means with unknown CVs.
Hence, it can be seen in this paper that the new estimator of Srivastava [12] is utilized and well established both in constructing the single mean confidence interval and the difference of means of normal distributions when the CVs are unknown.

Table 1 .
The coverage probabilities of 95% of the two-sided confidence intervals for the mean of the normal distribution with the unknown coefficient of variation (CV).

Table 2 .
The average lengths of 95% of two-sided confidence intervals for the mean of the normal distribution with unknown CV.

Table 3 .
The coverage probabilities of 95% of two-sided confidence intervals for the difference between the means of the normal distributions with unknown CVs.

Table 4 .
The average lengths of 95% of two-sided confidence intervals for the difference between the means of the normal distributions with unknown CVs.

Table 4 .
[9]t.The dataset, previously considered by Niwitpong[9], is fitted by the normal distribution.The data shows the cholesterol level of 15 participants who were given eight weeks of training to truly reduce the cholesterol level.The n = 15 participants were129, 131, 154, 172, 115, 126, 175, 191, 122, 238, 159, 156, 176, 175 and 126.The sample mean and sample variance of the data were 156.3333 and 1094.9520,respectively.The sample CV was 0.2117.The GCIs for the mean with unknown CV θ and θ * were, respectively, (136.8439,173.6214) and (137.9615,174.5876) with interval lengths of 36.7775 and 36.6261.The large sample confidence intervals for the mean with unknown CV θ and θ * were (−4679.6220,4991.3580) and (−4709.2810,5022.8840) with interval lengths of 9670.9800 and 9732.1650,respectively.Finally, the confidence interval for the mean µ based on the Student's t-distribution was (138.0087,174.6580) with an interval length of 36.6493.

Table 6 .
The coverage probability (average length) of 95% of two-sided confidence intervals for the mean of the normal distribution with unknown CV when n = 32, µ = 45.9688 and σ 2 = 27.7732.

Table 7 .
The coverage probability (average length) of 95% of two-sided confidence intervals for the difference between the means of the normal distributions with unknown CVs when n = 121, m = 75, µ X = 1.3000, µ Y = 4.1000, σ 2 X = 1.7040 and σ 2 Y = 4.0540.