Now consider a situation in which we compare the difference between means of samples selected from two populations.

*Z ***AND t STATISTICS FOR TWO INDEPENDENT SAMPLES**

Now consider a situation in which we compare the
difference between means of samples selected from two populations. In a
clinical trial, we could be comparing the mean of a variable (commonly referred
to as an endpoint) for a control group to the corresponding mean for a
treatment group.

First assume that both groups have normally
distributed observations with known and possibly different variances *σ** _{t}* and

Let us select two samples independently from the
two groups (treatment and control) and compute the means of the samples. We
denote the means of the samples from the control and treatment groups, * _{t }*and

which has a standard normal distribution. Here is
an interesting statistical observation: Even though we are finding the
difference between two sample means, the variance of the distribution of their
differences is equal to the sum of the two squared standard errors associated
with each of the individual sample means. The standard errors of the treatment
and control groups are calculated by dividing the population variance of each
group by the respective sample size of each independently selected sample.

As demonstrated in Section 8.6, the *Z* transformation, which employs the addition
of the error variances of the two means, enables us to obtain confidence
intervals for the difference between the means. In the special case where we
can assume that *σ*^{2}* _{t}* =

The term *σ*^{2} is referred to as the common variance. Since
P{[–1.96 ≤ Z ≤ 1.96] = 0.95, we find after algebraic manipulation that

is a 95% confidence interval for μ_{t} – μ_{c}.

In practice, the population variances of the treatment
and control groups are unknown; if the two variances can be assumed to be equal,
we can calculate an estimate of the common variance *σ*^{2}, called the pooled estimate. Let *S* ^{2}* _{t}* and

The corresponding *t* statistic is

This formula is obtained by replacing the common *s* in the formula above for *Z*
with the pooled estimate *S _{p}*.
The resulting statistic has Student’s

Although not covered in this text, the hypothesis
of equal variances can be tested by an *F*
test similar to the *F* tests that are
used in the analysis of variance (discussed in Chapter 13). If the *F* test indicates that the variances are
different, then one should use a statistic based on the assumption of unequal
variances.

This problem with unequal and unknown variances is
called the Behrens–Fisher problem. Let “*k*”
denote the test statistic that is commonly used in the Behrens– Fisher problem.
The test statistic *k* does not have a *t* distribution, but it can be
ap-proximated by a *t* distribution
with a degrees of freedom parameter that is not nec-essarily an integer. The statistic
*k* is obtained by replacing the *Z* statistic in the un-equal variance
case as given below:

where *S*^{2}* _{t}* and

We use a *t*
distribution with *v* degrees of freedom to approximate the distribution
of *k*. The degrees of freedom *v* are

This is the formula we use for confidence intervals
in Section 8.7 when the vari-ances are assumed to be unequal and also for
hypothesis testing under the same as-sumptions (not covered in the text).

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