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.