Question: Given the air-pollution data in Table 1.5, examine the pairs \(X_{5}=\mathrm{NO}_{2}\) and \(X_{6}=\mathrm{O}_{3}\) for bivariate normality.

1. Calculate statistical distances \(\left(\mathbf{x}_{j}-\overline{\mathbf{x}}\right)^{\prime} \mathbf{S}^{-1}\left(\mathbf{x}_{i}-\overline{\mathbf{x}}\right), j=1,2, \ldots, 42\), where \(\mathbf{x}_{j}^{\prime}=\left[x_{j 5}, x_{j 6}\right]\)
2. Determine the proportion of observations \(\mathbf{x}_{j}^{\prime}=\left[x_{j}, x_{16}\right], j=1,2, \ldots, 42\), falling within the approximate \(50 \%\) probability contour of a bivariate normal distribution.
3. Construct a chi-square plot of the ordered distances in Part a.

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