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 \ -0.181 & -0.125 & 0.740 \end{pmatrix} ] * Significance: (\alpha = 0.05), (F_{3,18}(0.05) = ____ ___ ___ _______ ___.
__________ ___ ____ ________ ________ _________ ___.
_______ __________ _________ ___ _________ _____ _______ __________.
_____ ______ __________ __________ _______.
______ ______ ________ _______ _____ ______ _______.
_______ _______ ________ _____ _______ __________.
_____ ___ ________ _______ _______ ________ ___ ______ ____ ________ ____ _____.
_____ _________ _____ ____ ____ ___ _____ ______ _________ ___ ___ ________.
______ _____ ___ _______ ________ _____ __________.
___ _________ ________ ___ ______ ______ ______ __________.
__________ ____ ___ __________ ________ __________ ___ __________ ______ ____ _______.
_______ ________ ________ ______ _______ _____ ____ ___ ________ ___.
______ ____ _____ ______ __________ ________ ______ ________ _________ _____ ____.
_________ ______ _______ _______ ________ _______.
___ _______ ___ ________ __________ ___ _________ _____.
_______ _____ _______ ___ ______ _________ _________ ______ ____ ______.
_____ ____ ________ _____ _______ ______ _______ _______ ________ ___ ____ __________.
___ __________ ______ _______ ________ _____ __________.
____ _______ ___ _________ ______ _____ _______ __________ ___ _________ _____ _____.
________ __________ ______ _________ ___ ____ _______ _______ _________.
_________ _______ ____ _____ ___ ___.
__________ _________ ____ _____ ________ ___ ________.
____ ________ _____ ____ ________.
________ ____ ____ _________ ________ _________ __________ ____ ______ _____.
__________ ________ ________ _______ __________ _________ _______ ____.
___ ___ _____ ___ _________ ___ _______ ______ _______ ______ _______.
_______ ____ ___ _____ _____ _________ ___ __________ _____ __________ _________ _____.
_____ _________ __________ _____ _________ _________ _____ _______ ______ ______ ___.
____ _________ __________ _________ _____ _________ _________ ________ ____ _______.
_________ ______ ________ _________ ___ ________ ____ _________ _______ ___.
________ __________ ____ ____ __________ ______ ______ _______ _____ ______ ___.
_________ _____ _______ _________ ___ __________.
_________ __________ ________ _________ _________ __________ ________ _____.
_____ ________ ________ ____ ________ _____ _____ _________ _________.
_____ _________ _________ ___ _________.
_____ ___ _________ ___ ____ ____ _____.
_______ ________ ____ ______ ___ ________ ___ _______ ________ ____ _________.
_________ ____ ___ ________ ____ _______ _____ ________.
______ __________ _____ ___ ___ ________ __________.
________ ________.
Get Full Answer on WhatsApp