Optimal confidence interval for the largest mean of correlated normal populations and its application to stock fund evaluation

A single-sample sampling procedure for obtaining an optimal confidence interval for the largest or smallest mean of several correlated normal populations is proposed. It is assumed that the common variance is either known or unknown and the common correlation coefficient is a given non-negative value. The optimal confidence interval is obtained by maximizing the coverage probability with the expected confidence width being fixed at a least favorable configuration of means. Statistical tables of the critical values are calculated to implement the optimal confidence interval. Finally, this interval procedure is employed to estimate the mean return of the best stock fund among four diversified mutual funds in the United States from the years of 1977 to 2005. It has been found, with 95% confidence, that the best mutual fund has a mean return falling between nine and twenty-one percent, and the worst one has a mean return falling in a range from three and fifteen percent. Therefore, it can be concluded that stock fund investment in the U.S. stock market outperforms the long term inflation rate.