7.2.8 - Simultaneous (1 - α) x 100% Confidence Intervals

As before, the next step is to determine how these notes differ. This may be carried out using the simultaneous \((1 - α) × 100\%\) confidence intervals.

For Large Samples: simultaneous \((1 - α) × 100\%\) confidence intervals may be calculated using the expression below:

\(\bar{x}_{1k} - \bar{x}_{2k} \pm \sqrt{\chi^2_{p,\alpha}\left(\dfrac{s^2_{1k}}{n_1} + \dfrac{s^2_{2k}}{n_2}\right)}\)

This involves the differences in the sample means for the kth variable, plus or minus the square root of the critical value from the chi-square table times the sum of the sample variances divided by their respective sample sizes.

For Small Samples: it is better to use the expression below:

\(\bar{x}_{1k} - \bar{x}_{2k} \pm \sqrt{\dfrac{p(n_1+n_2-2)}{n_1+n_2-p-1}F_{p,\nu,\alpha}}\sqrt{\left(\dfrac{s^2_{1k}}{n_1} + \dfrac{s^2_{2k}}{n_2}\right)}\)

Basically, the chi-square value and the square root are replaced by the critical value from the F-table, times a function of the number of variables, p, and the sample sizes n1 and n2.

Example 7-16: Swiss Bank Notes Section

An example of the large approximation for length is given by the hand calculation in the expression below:

\(214.969 - 214.823 \pm \sqrt{12.59 \times (\dfrac{0.15024}{100}+\dfrac{0.12401}{100})}\)

\((-0.040, 0.332)\)

Here the sample mean for the length of a genuine note was 214.969. We will subtract the sample mean for the length of a counterfeit note, 214.823. The critical value for a chi-square distribution with 6 degrees of freedom evaluated at 0.05 is 12.59. The sample variance for the first population of genuine notes is 0.15024 which we will divide by a sample size of 100. The sample variance for the second population of counterfeit notes is 0.12401 which will also divide by its sample size of 100. This yields a confidence interval that runs from -0.04 to 0.332.

The results of these calculations for each of the variables are summarized in the table below. Basically, they give us results that are comparable to the results we obtained earlier under the assumption of homogeneity for variance-covariance matrices.

Variable 95% Confidence Interval
Length -0.040, 0.332
Left Width -0.515, -0.199
Right Width -0.638, -0.308
Bottom Margin -2.687, -1.763
Top Margin -1.287, -0.643
Diagonal 1.813, 2.321