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:
\(\bar{x}_{1k} - \bar{x}_{2k} \pm t_{2n-2, \frac{\alpha}{2p}}\sqrt{\frac{2s^2_k}{n}}\)
\(214.959 - 214.813 \pm \underset{2.665}{\underbrace{t_{2\times 100-2, \frac{0.05}{2 \times 6}}}}\sqrt{\dfrac{2 \times 0.137}{100}}\)
\((0.006, 0.286)\)
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 combined 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 is 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:
Computing the 2 sample Bonferroni CIs
To compute the two-sample Bonferroni CIs:
- Open the ‘swiss3’ data set in a new worksheet
- Rename the columns type, length, left, right, bottom, top, and diag.
- Calc > Basic Statistics > 2-sample t
- Choose ‘Both samples are in one column’ in the first window.
- Highlight and select length to move it to the Samples window.
- Highlight and select type to move it to the Sample IDs window.
- Under ‘Options’, enter 99.17, which corresponds to 1-0.05/6, the adjusted individual confidence level for simultaneous 95% confidence with the Bonferroni method.
- Check the box to Assume equal variances.
- Select Difference not equal for the Alternative hypothesis.
- Select ‘OK’ twice. The intervals for length are displayed in the results area.
- Repeat steps 3. and 4. for each of the other 5 variables. The six resulting intervals together may be interpreted with simultaneous 95% confidence.
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 to 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.