5.4 Sample Size Determination 65mean difference will be less than the b that is determined from the
sample, and 5% of the time, it will be larger.
We conclude with 95% confi dence that the difference μ 1 − μ 2 is
less than b. If b < 0, we reject the notion that the group 1 mean is larger
than the group 2 mean, and conclude that group 2 has the larger mean.
We do not set a fi nite lower limit because we are not concerned about
how much larger it is. On the other hand, if we are only interested if
group 1 has a larger mean than group 2, we would take an interval of
the form [ a , ∞ ) and reject the notion that group 1 has a larger mean
than group 2 if a > 0. Here we do not worry about the upper bound
because we do not care how much larger it is.
In the next chapter, we will discuss hypothesis tests and will see
the relationship between hypothesis testing and confi dence intervals
presented there. The two - tailed and one - tailed hypothesis tests corre-
spond exactly to the two - sided and one - sided confi dence intervals.
We illustrated confi dence intervals for a one sample problem for
simplicity. This easily extends to the two sample situation for mean
differences and for other parameters in one - sample and two - sample
problems for parametric families of distributions. In our examples, Z
and T play the role of what we call pivotal quantities. A pivotal quantity
is a random variable whose distribution is known and the resulting
probability statement can be converted into a confi dence interval.
Because of the 1 – 1 correspondence between hypothesis testing and
confi dence intervals, nonparametric confi dence intervals can be
obtained through nonparametric tests. So too can bootstrap confi dence
interval be defi ned.
5.4 SAMPLE SIZE DETERMINATION
We will demonstrate fi xed sample size estimation criteria for confi -
dence intervals using parametric assumptions. The approach is to
specify a width or half - width for the interval and a confi dence level.
Then, the width can be expressed in terms of the sample size n. We
will demonstrate that for the estimation of a population mean and for
the difference between two population means.
Why is sample size determination important in medical research?
When conducting an experiment or a clinical trial, cost is an important
consideration. The number of tests in an experiment has an effect on the