The test statistic T2, under the hypothesis H, is distributed as Hotelling's T2 distribution with

parameters p and n1 + n2 2. We list below some important properties of the Hotelling's T2

statistic:

1. Hotelling's T2 distribution is skewed.

2. For a two-sided alternative hypothesis, the critical-region is one-tailed.

3. A necessary condition for the inverse of the pooled covariance matrix to exist is that the

n1 + n2 2 > p.

4. A straightforward, not necessarily simple, transformation of the Hotelling's statistic gives

us an F-statistic.

As in the previous section, we may also use the likelihood-ratio tests, which lead to an appropriate

2-test, for large n of course, see Rencher (2002) or Johnson and Wichern (2006). In

the next illustrative example, we obtain the Hotelling's test statistics, the associated F-statistic,

and the likelihood-ratio test.

1.and give interpretations of your results. What is a reasonable interpretation of this statistical test? Construct confidence regions for each problem to aide your interpretations. How might this process compare to testing each univariate variable independently and what problems might that incur?