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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.