The data in the table are from a growth experiment with blue-green algae Spirulina platensis conducted by Linda Shurtleff, North Carolina State University (data used with permission). These treatments...

The data in the table are from a growth experiment with blue-green algae Spirulina platensis conducted by Linda Shurtleff, North Carolina State University (data used with permission). These treatments were determined by the amount of “aeration” of the cultures:

1. no shaking and no CO₂
aeration;

2. CO₂
bubbled through the culture;

3. continuous shaking of the culture but no CO₂; and

4. CO₂
bubbled through the culture and continuous shaking of the culture. There were two replicates for each treatment, each consisting of 14 independent solutions. The 14 solutions in each replicate and treatment were randomly assigned for measurement to 1 of each of the 14 days of study. The dependent variable reported is a log-scale measurement of the increased absorbance of light by the solution, which is interpreted as a measure of algae density. The readings for DAY S = 0 are a constant zero and are to be omitted from the analyses.

(a) Use quadratic polynomials to represent the response over time. Fit a model that allows each treatment to have its own intercept and quadratic response. Then fit a model that allows each treatment to have its own intercept but forces all to have the same quadratic response. Use the results to test the homogeneity of the responses for the four treatments. (Note: Use the residual mean square from the analysis of variance as your estimate of σ².) Use the quadratic model you have adopted at this point and define a reduced model that will test the null hypothesis that all intercepts are zero. Complete the test and state your conclusions.

(b) The test of zero intercepts in Part (a) used quadratic polynomials. Repeat the test of zero intercepts using cubic polynomials for each treatment. Summarize the results.