An experiment was conducted at the University of North Carolina to see whether the biochemical oxygen demand (BOD) test for water pollution is biased by the presence of copper. In this test, the amount of dissolved oxygen in a sample of water is measured at the beginning and at the end of a five-day period; the difference in dissolved oxygen content is ascribed to the action of bacteria on the impurities in the sample and is called BOD. The question is whether dissolved copper retards the bacterial action and produces artificially low responses for the test.
The data in the following table are partial results from this experiment. The three samples (which are from different sources) are split into five subsamples, and the concentration of copper ions in each subsample is given. The BOD measurements are given for each subsample€”copper ion concentration combination.

a. Using dummy variables and treating the copper ion concentration as a nominal variable, provide an appropriate regression model for this experiment. Is this a randomized-blocks design?
b. If copper ion concentration is treated as an interval (continuous) variable, one appropriate regression model would be
Y = β0 + β1Z1 + β2Z2 + β3X + β4Z1X + β5Z2X + E
where

and X = copper ion concentration. What are the advantages and disadvantages of using the models in parts (a) and (b)? Which model would you prefer to use, and why?
c. Compare (without doing any statistical tests) the average BOD responses at the various levels of copper ion concentration.
d. Use the following table, which is based on a randomized-blocks analysis, to test (at a = .05) the null hypothesis that copper ion concentration has no effect on the BOD test.

e. Judging from the ANOVA table and the observed block means, does blocking appear to be justified?
f. The randomized-blocks analysis assumes that the relative differences in BOD responses at different levels of copper ion concentration are the same, regardless of the sample used; in other words, there is no copper ion concentration-sample interaction. One method (see Tukey 1949) for testing whether such an interaction effect actually exists is Tukey s test for additivity. It addresses the null hypothesis
H0: No interaction exists (i.e., the model is additive in the block and treatment effects).
versus the alternative hypothesis
HA: The model is not additive, and a transformation f(Y) exists that removes the nonadditivity in the model for Y.
Tukey's test statistic is given by

(which is distributed as F1, (k-1)(b-1)-1 under H0), where

using the notation in this chapter.
Given that the computed F statistic equals 4.37 in Tukey's test for additivity, is there significant evidence of nonadditivity? (Use α = .05.)
g. The following table gives results from fitting the multiple regression model given in part (b). Use this table to test whether evidence exists of a significant effect due to copper ion concentration (i.e., test H0: β3 = β4 = β5 = 0). Use the result that R2 = 0.888.

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Copper Ion Concentration (ppm) Sample 0.1 0.3 0.5 0.75 Mean 210 194 138 195 183 98 150 135 89 148 125 90 140 130 85 168.60 153.40 100.00 Mean 180.67 158.67 124.67 121.00 118.33 14067 1 for sample 1 z,#(0 for sample 2 -1 for sample 3 0 for sample 1 a-(1 for sample 2 -1 for sample 3 Source d.f MS Samples Concentrations Error 12,980.9333 9.196.6667 913.7333 6,490.4667 2.299.1667 114.2167 56.83 20.13 SSN SSE SSN)/Lk 1(b 1) 1] SSN = Source d.f MS 11,764.90 1,216.03 7,134.57 303.27 91.07 286.83 Zi Error