# Example

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Difference Confidence Intervals
Individual Conficence Intervals                                    Standard                     Paired Alpha =           0.05
Standard                              Individual Alpha =    0.05                              Mean      Error of                                           Mean
Group            Means               Deviations            n                Upper      Lower      Mean               Groups          Diff    Difference                     Upper     Lower Difference

1                  10                      24            64             15.87989 4.120108        10                 1 vs 2          9       4.242641                      17.31542 0.684577           9
2                  19                      24            64             24.87989 13.12011        19                 2 vs 3          9       4.242641                      17.31542 0.684577           9
3                  28                      24            64             33.87989 22.12011        28                 1 vs 3          18      4.242641                      26.31542 9.684577           18

Individual
Confidence Intervals                                                                                                                          Confidence Intervals
30
40
25
35

Difference Range
30                                                                                                                                                                        20
Mean Range

25
15
20
Assume that the mean of the second group is higher than the mean of the first group. Then the individual confidence intervals will overlap whenever:
15                                                                       
10
x 2  z 1 2  x1  z 1 1
10                                                         n2              n1
5
5                                                                                                                                                 1
2
2
2

While the confidence interval on the difference will indicate a significant difference when:                                           0  ( x 2  x 1 )  z 2        
n1       n2
0                                                                                                                                                                     0
1
2
2
2
1       2                                             1 vs 2
1
This means that overlapping will give the wrong conclusion whenever:2                                       z 2                   x 2  x1  z 1 (
3                       )
n1        n2                          n1       n2
Group
Since                       2
   2
1       2           2 1 2       and      z 2  z 1          the above interval will always exist.
1
       2
    (                 )2 
n1          n2               n1        n2           n1 n2

In contrast, if the individual confidence intervals do not overlap then the two means will be significantly different.

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