#### Title

Statistical Inference for a Normal Distribution with Variance as a Multiple of Its Mean

#### Date of Completion

January 2011

#### Keywords

Statistics

#### Degree

Ph.D.

#### Abstract

This dissertation addresses some interesting inferential problems for the mean of a special normal distribution whose variance is a multiple of its mean. It primarily focuses on determining the necessary sample size for tests of hypotheses when both error probabilities are pre-assigned. We develop both fixed-sample-size and sequential sampling methodologies which have been implemented and validated via simulations and real data. ^ We begin with fixed-sample-size test procedures and show how complications may arise due to a non-central distribution of the test statistic under both null and alternative hypotheses. We provide both exact and large-sample solutions to handle these issues. That way, our methodologies become readily implementable. In the fixed-sample case, we also briefly discuss some minimum variance unbiased estimation problems. ^ Next, we implement the full spectrum of Wald's (1947) *sequential probability ratio test* (SPRT) for the same testing problem. We have indicated how the customary sequential *t*-test in a one-sided case becomes simplified. The SPRT is immediately followed by Mukhopadhyay and de Silva's (2008) *random sequential probability ratio test* (RSPRT). For both SPRT and RSPRT, three appropriate truncation methods have been introduced and their features are contrasted. ^ Finally, we have successfully implemented our methodologies on interesting real datasets after statistical validation of the normality of these datasets. One illustration consists of data from a customary fixed-width confidence interval procedure. The other illustration uses a very interesting dataset on hourly 911 dispatches. Performances of both exact and large-sample methods are investigated and compared. ^ We conclude that our newly developed methodologies have worked remarkably well both theoretically as well as practically. The methodologies are rich enough to feed into a wide range of future research problems of importance. ^

#### Recommended Citation

Bhattacharjee, Debanjan, "Statistical Inference for a Normal Distribution with Variance as a Multiple of Its Mean" (2011). *Doctoral Dissertations*. AAI3485236.

https://opencommons.uconn.edu/dissertations/AAI3485236