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Chapter 13: Confidence Intervals – Small Samples
When we have independent samples, there is a procedure that we can use to
construct confidence intervals for the difference between two population
means when the sample sizes are small and the population variances or
standard deviations are unknown
.
Here the procedure assumes that the
samples are obtained from normal
populations
with the added assumption that the
populations have equal
variances or standard deviations.
Thus, assuming equal variances, we have that the distribution of the
differences of the sample means will have a standard deviation which can be
written as follows:
Since the standard deviation
is unknown, we can compute an estimate
for it. The estimate we use is called the pooled standard deviation,
denoted by
. The equation for the pooled variance is given below from
which we can compute the pooled standard deviation
.
Note:
The subscripts refer to the respective populations.
Thus we can write