IGNOU MST 18 2026 - 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) = 3.16) --- Step 1: Compute the difference between mean vectors [ \mathbf{d} = \bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_2 = \begin{pmatrix} 2\3\4 \end{pmatrix} -

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