NONCENTRAL T DISTRIBUTION: A REVIEW

Introduction

The noncentral t-distribution is a generalization of the central t-distribution. It is used to model the probability distribution of a random variable whose mean is different from the population mean. It is most commonly used in the analysis of experimental data, particularly in the fields of biostatistics and medical research. The noncentral t-distribution is also known as the non-standard t-distribution or the noncentral F-distribution.

Definition

The noncentral t-distribution is defined as the distribution of the statistic t* = (X-μ)/σ with mean μ and standard deviation σ, where X is a random variable with mean μ and variance σ2. The noncentral t-distribution is sometimes referred to as the noncentral F-distribution, which is the distribution of the ratio F* = (X-μ)/σ2.

Properties

The noncentral t-distribution is a continuous probability distribution, meaning that its probability density function (PDF) is continuous. The PDF of the noncentral t-distribution is characterized by its mean μ and variance σ2. The noncentral t-distribution is symmetric about its mean, and its PDF is unimodal. The noncentral t-distribution is also the distribution of the statistic t* = (X-μ)/σ, where X is a random variable with mean μ and variance σ2.

Applications

The noncentral t-distribution is used in a variety of applications. In biostatistics, for instance, it is used to assess the significance of an observed difference between two sample means. It is also used in medical research to test for differences in the means of two or more groups of patients. In addition, the noncentral t-distribution is used in the analysis of variance (ANOVA) to assess the significance of differences between groups.

Conclusion

The noncentral t-distribution is a generalization of the central t-distribution and is used to model the probability distribution of a random variable whose mean is different from the population mean. It is a continuous probability distribution, characterized by its mean μ and variance σ2. The noncentral t-distribution is used in a variety of applications, including biostatistics, medical research, and the analysis of variance (ANOVA).

References

Calderhead, B. (2015). Noncentral T Distribution. Retrieved from https://www.statisticshowto.datasciencecentral.com/noncentral-t-distribution/

Feller, W. (1968). An Introduction to Probability Theory and Its Applications. New York: John Wiley & Sons.

O’Hagan, A., & West, M. (2007). Noncentral t-Distribution. In Encyclopedia of Statistics in Behavioral Science (pp. 1477–1479). Hoboken, NJ: John Wiley & Sons.

Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Belmont, CA: Thomson Brooks/Cole.