A random sample of 12 factories was conducted for the pairs of observations on sales (X₁) and demands ( X₂)  and the following information was obtained:

The expected mean vector and variance covariance matrix for the factories in the population are:

=

and

Test whether the sample confirms its truthness of mean vector at 5% level of
significance, if:
i) Σ is known,
ii) Σ is unknown.

To test the hypothesis regarding the mean vector of the population based on a random sample, we can use the Hotelling's T-squared test. The test statistic is given by:

\[ T^2 = n(\bar{X} - \mu)^T \Sigma^{-1} (\bar{X} - \mu) \]

where:
- \( n \) is the sample size (12 in this case),
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