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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

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 |

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

*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.*

**alphabetic order!**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.