Let two independent samples of size and
from trivariate normal populations have following mean vectors and variance-covariance matrices:
and
and
, respectively.
---$
$
samples are significantly different from each other at 5% level of significance or not.$
We are asked to test whether the mean vectors of two independent multivariate normal samples are significantly different using Hotelling’s (T^2) test. Let's solve it numerically step by step. --- Given: * Sample sizes: (N_1 = 10), (N_2 = 12) * Mean vectors: (\bar{\mathbf{x}}_1 = \begin{pmatrix}2\3\4\end{pmatrix}, \quad \bar{\mathbf{x}}_2 = \begin{pmatrix}5\6\7\end{pmatrix}) * Pooled covariance inverse: [ \mathbf{S}_p^{-1} = \begin{pmatrix} 1.138 & -0.152 & -0.181 \ -0.152 & 0.837 & -0.125 \ -0.181 & -0.125 & 0.740 \end{pmatrix} ] * Significance: (\alpha = 0.05), (F_{3,18}(0.05) = _______ ___ ___ ___ _________ ______ ______ ____ ___ ________ ________.
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Let two independent samples of size and
from trivariate normal populations have following mean vectors and variance-covariance matrices:
and
and
, respectively.
---$
$
samples are significantly different from each other at 5% level of significance or not.$
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
Let , where
and
.
Then find and
.
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
Obtain the square root matrix corresponding to a matrix . Also
verify that .
State whether the following statements are True or False. Give reasons in support of your answer: 5×2=10
(a) The eigenvalues of a positive definite matrix are less than zero.
(b) is an orthogonal and idempotent matrix.
(c) A factor model postulates that a random vector is linearly dependent upon a few observable common factors.
(d) A given px1 vector is a plausible value for the mean of a multivariate normal distribution can be tested using Student's t test.
(e) In single linkage method, we find the minimum distance in the distance matrix and merge the minimum distanced clusters