### Abstract:

In the problem of testing the median using a random sample from a
certain distribution, and if no other parametric family is suggested,
the t-test is known to be the optimal procedure when this distribution
is normal. If the sample appears to be non-normal, one has the choice
either to consider a non-parametric approach or to try to correct for
non-normality before applying the t-test.
In this thesis we investigate the effect of applying certain power
transformations as an action to correct for non-normality before
applying the t-test. Also we investigate the effect of applying a
power transformation then trimming a certain proportion from the data
on each tail as a double action to correct for non-normality. This
problem is first considered by Doksum and Wong (1983), who apply the
Box-Cox power transformations to positive, right-skewed data when
testing for the equality of distributions of two independent samples.
In the present work we provide results for the one-sample case
using two alternatives to the Box-Cox power family which are applicable
to all data sets. Whenever it can be assumed that the data is a random
sample from a symmetric distribution with heavy tails, it is shown that
the John-Draper family of modtlus power transformations, with the
transformation parameter being positive and smaller than 1 , is
appropriate to correct for non-normality and the t-test based on the
transformed data is asymptotically more efficient and has better power
properties than the t-test based on the data in its original scale.
-When the data is thought to have a skewed distribution and can assume
negative as well as positive values, a new family of transformations,
referred to as the two-domain family, is introduced. It is shown that
the t-test based on the data after applying this new transformation is
also asymptotically more efficient and has better power properties than
the t-test in the original scale. A simulation study shows that
trimming a certain proportion on each tail of the data transformed by
one of the above two transformations then applying the t-test to the
trimmed samples yields a considerable gain in power compared to the
t-test in the original scale.