7.2.3 - Example: Swiss Bank Notes

Example 7-15: Swiss Bank Notes Section

An example of the calculation of simultaneous confidence intervals using the Swiss Bank Notes data is given in the expression below:

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

Here we note that the sample sizes are both equal to 100, \(n = n_{1} = n_{2} =100\), so there is going to be simplification of our formula inside the radicals as shown above.

Carrying out the math for the variable Length, we end up with an interval that runs from -0.044 to 0.336 as shown below.

Using SAS

The SAS program, below, can be used to compute the simultaneous confidence intervals for the 6 variables.

Download the SAS program here: swiss11.sas

View the video explanation of the SAS code.

The downloadable results as listed here: swiss11.lst.

Using Minitab

At this time Minitab does not support this procedure.

Analysis

Confidence Intervals - Swiss Bank Notes
Obs Variable type _TYPE _FREQ_ n1 xbar1 s21 n2 xbar1 s22
1 bottom fake 0 100 100 8.305 0.41321 100 10.530 1.28131
2 diagon fake 0 100 100 141.517 0.19981 100 139.450 0.31121
3 left fake 0 100 100 129.943 0.13258 100 130.300 0.06505
4 length fake 0 100 100 214.969 0.15024 100 214.823 0.12401
5 right fake 0 100 100 129.720 0.12626 100 130.193 0.08894
6 top fake 0 100 100 10.168 0.42119 100 11.133 0.40446

Confidence Intervals - Swiss Bank Notes
Obs f t sp losim upsim lobon upbon
1 2.14580 2.66503 0.84726 -2.69809 1.75191 -2.57192 -1.97808
2 2.14580 2.66503 0.2551 1.80720 2.32680 1.87649 2.25751
3 2.14580 2.66503 0.09881 -0.51857 -0.19543 -0.47547 -0.23853
4 2.14580 2.66503 0.13713 -0.04433 0.33633 0.00643 0.28557
5 2.14580 2.66503 0.10760 -0.64160 -0.30440 -0.59663 -0.34937
6 2.14580 2.66503

0.41282

 -1.29523 -0.63477 -1.20716 -0.72284
 

Thebounds of the simultaneous confidence intervals are given in columns for losim and upsim. Those entries are copied into the table below:

Variable 95% Confidence Interval
Length -0.044, 0.336
Left Width -0.519, -0.195
Right Width -0.642, -0.304
Bottom Margin -2.698, -1.752
Top Margin -1.295, -0.635
Diagonal 1.807, 2.327

You need to be careful where they appear in the table in the output.

Note! The variables are now sorted in alphabetic order! For example, length would be the fourth line of the output data. In any case you should be able to find the numbers for the lower and upper bound of the simultaneous confidence intervals from the SAS output and see where they appear in the table above. The interval for length, for example, can then be seen to be -0.044 to 0.336 as was obtained from the hand calculations previously.

When interpreting these intervals we need to see which intervals include 0, which ones fall entirely below 0, and which ones fall entirely above 0.

The first thing that we notice is that interval for length includes 0. This suggests that we can not distinguish between the lengths of the counterfeit and genuine bank notes. The intervals for both width measurements fall below 0.

Since these intervals are being calculated by taking the genuine notes minus the counterfeit notes this would suggest that the counterfeit notes are larger on these variables and we can conclude that the left and right margins of the counterfeit notes are wider than the genuine notes.

Similarly we can conclude that the top and bottom margins of the counterfeit are also too large. Note, however, that the interval for the diagonal measurements fall entirely above 0. This suggests that the diagonal measurements of the counterfeit notes are smaller than that of the genuine notes.

Conclusions

  • Counterfeit notes are too wide on both the left and right margins.
  • The top and bottom margins of the counterfeit notes are too large.
  • The diagonal measurement of the counterfeit notes is smaller than that of the genuine notes.
  • Cannot distinguish between the lengths of the counterfeit and genuine bank notes.