Download nP: Number of data points (= sample size = degrees of freedom) for

nP : Number of data points (= sample size = degrees of freedom) for “present”
x̄P : Empirical mean for “present”
sP : Empirical standard deviation for “present”
nF , x̄F , sF : Same for “future”
assume sF = sP (same statistics in future).
t-test as in textbook: test quantity
x̄F − x̄P
u= p
(nP − 1)s2P + (nF − 1)s2F
s
nP nF (nP + nF − 2)
nP + nF
(1)
is t-distributed with f = nP + nF − 2 degrees of freedom. The mean values x̄P and x̄F differ
significantly if
u ≥ t0.05;f
(2)
(quantile of the t-distribution, one-sided). Substituting u, dividing by x̄P and rearranging, the condition becomes: The reduction ̺ is significant if
r
x̄F − x̄P
nP + nF sP
̺ :=
≥ t0.05;f
·
(3)
x̄P
nP nF
x̄P
For equal sampling size (nF = nP = n), things simplify to
r
2 sP
·
̺ ≥ t0.05;f
n x̄P
(4)
Relevant cases:
q
n f t0.05;f t0.05;f n2
5 8
1.86
1.176
10 18
1.73
0.774
20 38
1.68
0.531
40 78
1.67
0.373
for
sP
x̄P
= 0.41ppm
for
1.31ppm
̺ ≥ 37%
̺ ≥ 24%
̺ ≥ 16%
̺ ≥ 12%
sP
x̄P
0.91ppm
= 11.05ppm
̺ ≥ 9.7%
̺ ≥ 6.4%
̺ ≥ 4.4%
̺ ≥ 3.1%
For large sample size, the t-distribution becomes the normal distribution. Written in the form
“error of the mean divided by mean” the condition gets
√
√ sP / n
̺ ≥ t0.05;∞ 2 ·
(5)
| {z } x̄P
2.32
1