Please Note that the bold Red sections are to be included in the Data Analysis Benchmark
Research Question: Does the absorption of CO2 alter the pH of sea water to the point where damage is
caused to coral and other marine organisms?
Does the increase in the pressure of CO2 increase the rate of change of pH in seawater?
H01: Assuming a constant partial pressure of CO2, there is no change over time in the pH of the water
(due to the loss of CaCO3 in seashells that have been introduced into the system).
H02: There is no change in the pH of the water as the partial pressure of CO2 is increased
COs - I
0.03 kPa
COs -II
0.06 kPa
COs -III
0.12 kPa
7.9
7.7
7.7
7.8
7.7
7.7
7.9
7.9
7.8
Change in pH/24 hour interals
7.8
7.2
6.8
6.7
7.7
7.4
7.5
6.8
7.6
7.1
6.9
6.8
7.4
6.7
6.4
6.2
7.2
6.8
6.3
6.0
7.3
6.9
6.2
6.0
7.0
6.4
6.0
5.8
6.9
6.2
6.0
5.6
6.9
6.2
5.9
5.8
6.6
6.4
6.7
5.8
5.7
5.8
5.2
5.2
5.1
Dixon's Q-test: Detection of a single outlier
What if you encounter a recorded value that you believe is too removed the body of your other values?
7.5 − 6.9
= 0.88
7.5 − 6.8
Since 0.88 < 0.945 (CL90%), the value 0.88 cannot be dropped
Mean and Standard Deviation
COs - I
COs -II
COs -III
0.03 kPa mean
S.D
0.06 kPa mean
S.D
0.12 kPa mean
S.D
7.8
0.12
7.7
0.06
7.9
0.06
7.7
0.10
7.3
0.10
7.0
0.06
Change in pH
7.2
7.1
0.15
0.14
6.8
6.3
0.10
0.10
6.4
6.0
6.2
0.06
6.8
0.05
6.1
0.11
5.7
0.12
6.6
0.15
5.8
0.06
5.2
.06
Now before we decide to check the null hypotheses let’s take a look at the graphs and see if we can find
any meaningful patterns.
H0: Assuming a constant partial pressure of CO2, there is no change over time in the pH of the water
(due to the loss of CaCO3 in seashells that have been introduced into the system).
The above data shows that in each case, there was a change in pH over a 144 hour interval of time. The
change appears to be exponential with a low enough RMSE in each case to suggest that the trends vary
with the pressure of the COs applied over the surface of the water,
The average rate of change appears to increase as the partial pressure of the CO2 increases. The question
we need to ask is whether that change relates to the differences between the partial pressure of CO2 in
each of the systems. This question is addressed by the second null hypotheses:
H02: There is no change in the pH of the water as the partial pressure of CO2 is increased
Taking the tangents of each of the curves gives is the average slopes which translate out to change in
pH/time interval. This represents the rate of change in each of the systems (eg. Miles per hour)
Test Sample
Partial Pressure of CO2
Average Rate of Change RMSE
in pH/interval
COs - I
0.03 kPa
.010
.092
COs -II
0.06 kPa
.016
0.11
COs -III
0.12 kPa
.021
0.20
These date suggest that the rate of change in pH increases as the partial pressure of the CO s increases.
The RMSE values are small enough to support the idea that these values might represent what is really
going on. If I graphed those differences, the argument might appear to be even more convincing.
The rate of change in the pH appears to be increasing logarithmically (RMSE = .00004) as the partial
pressure of CO2 increases. The question is, are the differences in the rate of change at the three difference
partial pressures are real.
Generally these questions are addressed using inferential analysis; a technique that allows us to
determine if the differences between groups of data are statistically significant. The more common tests
used include the T-test, Chi Square test, and Analysis of Variance (ANOVA). All three tests are simple to
use. One must establish, however, which form of analysis would be more appropriate. And that depends
upon the conditions of the study.
Because we are dealing with three groups of data (the differing partial pressures of each of the systems)
and because of other considerations as well, we will use the ANOVA.
The results of an ANOVA statistical test performed at 10:51 on 2-FEB-2015 on the above data:
Source of
Variation
Sum of
Squares
between 2.138
error
8.527
total
10.66
d.f. Mean
Squares
2 1.069
15 0.5684
17
F
1.880
The probability of this result, assuming the null hypothesis, is 0.187
Group A: Number of items= 6
6.60 6.80 7.10 7.20 7.70 7.80
Mean = 7.20
95% confidence interval for Mean: 6.544 thru 7.856
Standard Deviation = 0.477
Hi = 7.80 Low = 6.60
Group B: Number of items= 6
5.80 6.10 6.30 6.80 7.30 7.70
Mean = 6.67
95% confidence interval for Mean: 6.011 thru 7.323
Standard Deviation = 0.734
Hi = 7.70 Low = 5.80
Group C: Number of items= 6
5.20 5.70 6.00 6.40 7.00 7.90
Mean = 6.37
95% confidence interval for Mean: 5.711 thru 7.023
Standard Deviation = 0.969
Hi = 7.90 Low = 5.20
Now let’s assume for a moment that I had decided to apply only two different partial pressures and then
asked the same question: is there a significant difference between the means of the two sets of data?
COs - I
COs -II
COs -III
0.03 kPa mean
S.D
0.06 kPa mean
S.D
0.12 kPa mean
S.D
7.8
0.12
7.7
0.06
7.9
0.06
7.7
0.10
7.3
0.10
7.0
0.06
Change in pH
7.2
7.1
0.15
0.14
6.8
6.3
0.10
0.10
6.4
6.0
6.2
0.06
6.8
0.05
6.1
0.11
5.7
0.12
6.6
0.15
5.8
0.06
5.2
.06
In this case I would probably not use an ANOVA. Most often when one is looking at two different groups
a t-test is used to establish whether there is a meaningful difference between their means. And I will
choose a one-tailed T-test because I’m only attempting to establish that the difference occurs in one
direction; this is, the pH is going to decline
This is what the results of the analysis would look like this: The T-value is 1.889833. The P-value is
0.044044. The result is significant at p< 0.05 so we can reject the second null hypothesis. Now I suppose
that under a different set of circumstances we might have looked at the first two data sets.
COs - I
COs -II
COs -III
0.03 kPa mean
S.D
0.06 kPa mean
S.D
0.12 kPa mean
S.D
7.8
0.12
7.7
0.06
7.9
0.06
7.7
0.10
7.3
0.10
7.0
0.06
Change in pH
7.2
7.1
0.15
0.14
6.8
6.3
0.10
0.10
6.4
6.0
6.2
0.06
6.8
0.05
6.1
0.11
5.7
0.12
6.6
0.15
5.8
0.06
5.2
.06
Here, the P value happens to be larger than the p <0.05 value (0.083278). Under these circumstances we
would have been unable to reject the second hypothesis. In any case, I would have never gone for two
sets, only.