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

Warisa Thangjai,Suparat Niwitpong,Sa-Aat Niwitpong

doi:10.3390/math5030039

Warisa Thangjai, Suparat Niwitpong + Show 1 more

Open Access

https://doi.org/10.3390/math5030039

Copy DOI

Abstract

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.

Highlights

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]
This paper extends the work of Sodanin et al [16] to construct confidence intervals for the normal population mean with unknown coefficient of variation (CV) based on the generalized confidence interval (GCI)
This paper provides generalized confidence intervals (CIGCI.θ and CIGCI.θ∗ ) and proposes large sample confidence intervals (CILS.θ and CILS.θ∗ ) for the single mean of the normal distribution with unknown

Summary

Introduction

It is well known that the sample mean, x, is the uniformly minimum variance unbiased (UMVU). Three new confidence intervals for the difference between normal means with unknown CVs were 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. 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: CIGCI.δ = (Rδ (α/2) , Rδ (1 − α/2)). 2 + σ2 2 mμ 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: CILS.δ =. 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: θX − θY −. For CIGCI.δ∗ , CILS.δ∗ and CIMOVER.δ∗ , the fraction of times that all δ∗ are in their corresponding confidence intervals provides an estimate of the coverage probability

Simulation Studies

An Empirical Application

Discussion and Conclusions