As in the one-sample case, the simultaneous confidence intervals should be computed only when we are interested in linear combinations of the variables. If the only intervals of interest, however, are the confidence intervals for the individual variables with no linear combinations, then a better approach is to calculate the Bonferroni corrected confidence intervals as given in the expression below:
\(\bar{x}_{1k} - \bar{x}_{2k} \pm t_{n_1+n_2-2, \frac{\alpha}{2p}}\sqrt{s^2_k\left(\dfrac{1}{n_1}+\frac{1}{n_2}\right)}\)
Carrying out the math we end up with an interval that runs from 0.006 to 0.286 as shown below:
\(\bar{x}_{1k} - \bar{x}_{2k} \pm t_{2n-2, \frac{\alpha}{2p}}\sqrt{\frac{2s^2_k}{n}}\)
\(214.959 - 214.813 \pm \underset{2.665}{\underbrace{t_{2\times 100-2, \frac{0.05}{2 \times 6}}}}\sqrt{\dfrac{2 \times 0.137}{100}}\)
\((0.006, 0.286)\)
Using SAS
These calculations can also be obtained from the SAS program as shown below:
Download the SAS Program here: swiss11.sas
Looking at the data step combine and moving down, we can see that the fourth line sets t=tinv. This calculates the critical value from the t-table as desired. Then the lower and upper bounds for the Bonferroni intervals are calculated under lobon and upbon at the bottom of this data step.
The downloadable output as given here: swiss11.lst, places the results in the columns for lobon and upbon.
Again, make sure you note that the variables are given in alphabetical order rather than in the original order of the data. In any case, you should be able to see where the numbers in the SAS output appear in the table below:
Using Minitab
At this time Minitab does not support this procedure.
Analysis
In summary, we have:
| Variable | 95% Simultaneous Confidence Intervals (Bonferroni corrected) |
|---|---|
| Length | 0.006, 0.286 |
| Left Width | -0.475, -0.239 |
| Right Width | -0.597, -0.349 |
| Bottom Margin | -2.572, -1.878 |
| Top Margin | -1.207, -0.723 |
| Diagonal | 1.876, 2.258 |
The intervals are interpreted in a way similar as before. Here we can see that:
Conclusions
- Length: Genuine notes are longer than counterfeit notes.
- Left-width and Right-width: Counterfeit notes are too wide on both the left and right margins.
- Top and Bottom margins: Counterfeit notes are too large.
- Diagonal measurement: The counterfeit notes are smaller than the genuine notes.