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Multivariate Analyses |

The variable is distributed
as a bivariate normal variate with mean 0 and covariance
,and it is independent of **S**.
The confidence ellipse for is based on
Hotelling's *T ^{2}* statistic:

A confidence ellipse for prediction is a confidence region
for predicting a new observation in the population.
It also approximates a region containing
a specified percentage of the population.
Consider **Z** as a bivariate random
variable for a new observation.
The variable is distributed
as a bivariate normal variate with mean 0 and
covariance ,and it is independent of **S**.

A confidence ellipse for prediction is then given by the equation

The family of ellipses generated by different
*F* critical values has a common center (the
sample mean) and common major and minor axes.
The ellipses graphically indicate the
correlation between two variables.
When the variable axes are standardized (by dividing the
variables by their respective standard deviations), the ratio
of the two axis lengths (in Euclidean distances) reflects the
magnitude of the correlation between the two variables.
A ratio of 1 between the major and minor axes
corresponds to a circular confidence contour
and indicates that the variables are uncorrelated.
A larger value of the ratio indicates a larger positive
or negative correlation between the variables.

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