# Statistics for Environmental Engineers

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Tukey (1949) showed that the confidence interval for the difference in two means (n and nj), taking into account that all possible comparisons of k treatments may be made, is given by:

yi— yj ±

qk,v,—/2

72

spool

J

1 1

n + nj

where qk,v,a/2 is the upper significance level of the studentized range for k means and v degrees of freedom in the estimate spool of the variance a2. This formula is exact if the numbers of observations in all the averages are equal, and approximate if the k treatments have different numbers of observations. The value of sp2ool is obtained by pooling sample variances over all k treatments:

2    = (n — 1)s2+ ••• +(nt1)s2

spod    n 1 + • • • + n k — k

The size of the confidence interval is larger when qk,v,a/2 is used than for the t statistic. This is because the studentized range allows for the possibility that any one of the k(k — 1) / 2 possible pair-wise comparisons might be selected for the test. Critical values of qk,v,a/2 have been tabulated by Harter (1960) and may be found in the statistical tables of Rohlf and Sokal (1981) and Pearson and Hartley (1966). Table 20.3 gives a few values for computing the two-sided 95% confidence interval.

### Solution: Tukey’s Method

For this example, k = 5, spool = 0.51, spool = 0.71, v = 50 — 5 = 45, and q5,40,0.05/2 = 4.49. This gives the 95% confidence limits of:

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