Let two independent samples of size and from tri

Question

Let two independent samples of size $\text{N}_{1} = 10$ and $\text{N}_{2} = 12$ from trivariate normal populations have following mean vectors and variance-covariance matrices:
$\bar{\text{x}}_{1} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ and $\text{S}_{1} = \begin{pmatrix} 1.2 & 0.3 & 0.2 \\ 0.3 & 1.5 & 0.4 \\ 0.2 & 0.4 & 1.8 \end{pmatrix}$
$\bar{\text{x}}_{2} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}$ and $\text{S}_{2} = \begin{pmatrix} 1.0 & 0.2 & 0.3 \\ 0.2 & 1.4 & 0.3 \\ 0.3 & 0.3 & 1.6 \end{pmatrix}$ , respectively.
---
$$If \mathbf{S}_p^{-1} = \begin{pmatrix} 1.138 & -0.152 & -0.181 \\ -0.152 & 0.837 & -0.125 \\ -0.181 & -0.125 & 0.740 \end{pmatrix} \text{is the inverse of the pooled covariance matrix}$ $
$$\mathbf{S}_p = \begin{pmatrix} 1.09 & 0.245 & 0.255 \\ 0.245 & 1.445 & 0.345 \\ 0.255 & 0.345 & 1.69 \end{pmatrix}, \text{then test whether the mean vectors of the two}$ $
samples are significantly different from each other at 5% level of significance or not.
$$\text{(Use } F_{3, 18}(0.05) = 3.16 \text{ )}$ $

24 Mar 2026

Answer :

Word Count : 421

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 \ ___ ________ ____ _____ _____ ______ _________.
_________ ___ _____ _________ _________ ________.
________ ___ _______ ____ ___ _________ _________.
_________ _________ _________ ___ _____ ___ ______ __________ _____ ___ _____ _______.
_____ _____ ______ ___ __________ ____ ________ _______ ____ ___.
__________ _____ ______ __________ ___ ____ ___ __________ ______ ______.
_________ __________ ________ _______ _____ ______ ____ _____.
___ _____ ________ ___ __________ _____ ______ _________ _______.
_________ _______ _________ _________ _____ ________ _________ __________.
________ ____ _____ ________ ________ _________ ________ _____ _______.
_________ ______ __________ __________ _____ _______.
_________ ______ _______ _________ ____.
_________ ________ ________ ______ _____ ___ ________ ____.
_______ ___ ____ _________ ______ _________ _____ __________ ______.
___ ________ ________ _____ __________ __________ _____ ________ ________ _____ ________ ____.
______ __________ ______ _____ _________ _______ ____ ____ _______ _____.
__________ _________ _______ ________ _______.
__________ ___ __________ ___ ________ _________ _________ ____ _________.
__________ ______ ____ ________ _____ __________ ________ ___ _________.
__________ __________ ________ _______ ________ ____ __________ ___ _______ ___.
_______ ___ ____ ________ ______ _________ ____ _________ ____ ______ ____.
____ __________ _____ __________ ______ ________ ________ _______ ____ _______ _________ __________.
_____ __________ ____ __________ ____ __________ ______ _____ _________ _________.
_____ __________ __________ ________ _______ _______ _______.
________ _______ _______ ______ _____ ________ __________ ___ _______ ________ _____ ____.
_________ ______ ______ ______ _________ _________ ___ ___ _________ _____ ____ ____.
_______ ______ ______ _____ __________.
______ ___ ____ _____ ______ _____.
__________ _____ ____ ___ ______.
_____ ______ __________ _______ _____.
________ ___ _______ __________ ____ ___.
__________ _______ __________ _____ ________.
____ _______ _________ ________ ______.
__________ ______ __________ ______ ______ __________ ___ _________ _____ ______.
________ ___ __________ __________ _____ _______.
__________ ________ _____ ____ _________ _______.
______ ____ ______ _________ __________ _____ _____ ________ _______ ________ _______.
____ _________ _________ ______ __________ _____ _______ ______ _____ __________.
__________ __________ ___ ______ ____.
________ __________ _____ ____ __________ _____ _______ ________ _________ _________.
___ ____ _______ _________ _______ ________ ________ _________ _________.
__________ _______ ____ _________ ______ _____ ___.
___ _________ _______ _____ ______ ______ ___.
Get Full Answer on WhatsApp