The "Relationship" column was created by comparing the "T value" column (t0) with values
from a Students' t table using the degrees of freedom in the "d.f." column. First, we tested to
see if the difference was zero. If this was not rejected, we labeled the "Relationship" column
"Accept Null Hypoth."
If the null hypothesis was rejected, Equations 7 and 8 were used with the appropriate values
from the Students' t table to determine whether the transect data were greater or lesser than the
fixed monitor station (FMS) data. These results were labeled in the "Relationship" column as
"Transect > FMS" or "FMS > Transect," respectively.
For 11 of the 23 stations, the statistical tests rejected the hypothesis that the FMS and transect
data were equal. This means that data collected at these FMS sites did not reflect the water
quality conditions occurring across the river.
These stations, which had nonequivalent FMS and transect comparisons, are marked with an
ampersand. It is recommended that further analysis be conducted on these stations to determine
if the fixed monitor system needs to be moved, modified, or increased in scope. It is possible
that the differences detected occur uniformly, allowing a simple addition or subtraction from the
FMS data to then accurately represent river conditions. If the variance is large temporally or
spatially, these stations should be relocated. To ensure the validity of these conclusions, it is
generally accepted that a sample size of at least seven is necessary.
At the remaining 12 stations, the null hypothesis that the FMS and transect data were equal
was accepted. This may be because the FMS adequately represents the transect, or simply
because the limited data did not provide sufficient evidence to reject the null hypothesis. Thus,
further analysis is needed to determine whether the monitors represent the flow.
Determining the Power of the Test
Using Equation 9, we calculated the statistic d for each station where the null hypothesis was
accepted. These results are shown in Table 4. The table shows that in no case is the power
greater than 0.32. Thus, we conclude that in each case where the conclusion of the test was to
accept the null hypothesis (fixed monitor data represents water quality conditions in the river),
there are insufficient data to make a reasonable statistical decision.
We next calculated the necessary sample size for each of these 12 stations to obtain the
desired target power of 0.8. These values are shown in Table 5. With the exception of stations
KLAW and MCQO, the sample sizes are somewhat unrealistic. This occurs because of the
relationships between the sample means and standard deviations.
From Equation 9, d =
. The power of the test relies on this relationship, in addition to the
sample size n . In these other stations, the variance is so large compared with the mean that
sample sizes are not reasonable. This implies that the fixed monitors are not located in such a
way that their values change uniformly with the flow values. Thus, a first step at improving
these monitors would be to place them in locations experiencing more uniform changes with
flow and to increase the number of fixed monitor locations across the flow.
Water Quality Technical Note AM-03 (January 1998)