analysis of a pilot-scale anaerobic baffled reactor treating domestic ...

that the means are the same, or conversely that at the 95% confidence level, the hypothesis that the
means are not the same cannot be rejected.
4.4 Ratio of two means: Fieller’s theorem
Fieller’s theorem states that the confidence limit of a/b, the ratio of two means a and b each with
variance V(a) and V(b) is (Davies and Goldsmith, 1977 p. 236):
a
b
Where t is the appropriate t-statistic at a significance level α and for Øn degrees of freedom. C(a,b) is
the co-variance of variables a and b. Where a and b are independently determined (i.e. calculated from
different experimental data sets) then C(a,b) = 0 and (4) reduces to
a
b
4.5 Linear Regression
Linear regressions can easily be performed by a variety of Microsoft Office Excel functions. The
method for calculating a least squares slope to describe a linear relationship between two data sets is
presented here since several of the quantities calculated in the process of calculating the least squares
slope are used in determining confidence intervals of the regression coefficients and of values
predicted by the regression (Davies and Goldsmith, 1977 pp. 185-206). For the independent data
points xi (e.g. time), and dependent data points yi (e.g. concentration data for times xi):
Sxx is the sum of squares about the mean of the independent data set:
S
S
S
2
t C
−
b
xx
yy
xy
±
=
=
t
b
∑
i
∑
i
⎛
⎜
∑
⎞
x ⎟
i
2 ( ) ⎝ i
x −
⎠
− ( x − x)
i
⎛
⎜
n
∑
2
∑
i
2 ( ) ⎝ i
y −
⎠
− ( y − y)
i
⎞
y ⎟
⎛ ⎞⎛
⎜∑
xi
⎟⎜
=
⎝ ⎠⎝
∑ i i
n
i
( a,
b)
2
V
( a)
±
t
b
V
n
( a)
− C(
a,
b)
2
2a
b
2
2
a t V
+ V ( b)
−
2
b
2
t V ( b)
1−
2
b
i
∑
i
i
( x ⋅ y ) −
−
i
∑
i
2
t V
1 −
b
( b)
⋅V
( a)
Syy is the sum of squares about the mean of the dependent data set:
Sxy is the sum of the product about the means:
b
i
2
⎞
y i ⎟
⎠
2
2
2
a t V
+ V ( b)
−
2
2
b b
2
( b)
2
189
( b)
⎛
⎜
V
⎝
( a)
C
−
( a,
b)
V ( b)
2
⎞
⎟
⎠
(4)
(5)
(6)
(7)