The following experiment is from Kuehl (2000). Sludge is a dried product remaining from processed sewage; it contains nutrients beneficial to plant growth. It can be used for fertilizer on agricultural crops provided it does not contain toxic levels of certain elements such as heavy metals (such as zinc, not rock groups). Typically, the levels of metals in sludge are assayed by growing plants in media containing different doses of the sludge.
A soil scientist hypothesized the concentration of certain heavy metals in sludge would differ among the metropolitan areas from which the sludge was obtained. The variation could result from any number of reasons, including the different industrial bases surrounding the areas and the efficiency of the various sewage treatment facilities. If this was true, then recommendations for applications on crops would have to be preceded by knowledge about the source of the sludge material. An assay was planned to determine whether there was significant variation in heavy metal concentrations among diverse metropolitan areas.
The investigator obtained sewage sludge from treatment plants located in three different metropolitan areas. Barley plants were grown in a sand medium to which sludge was added as fertilizer. The sludge was added to the sand at three different rates: 0.5, 1.0, and 1.5 metric tons / acre. Each of the nine treatment combinations was randomly assigned to four replicate containers. The containers were arranged completely at random in a growth chamber. At a certain stage of growth, the zinc contents in parts per million were determined for the barley plants grown in each of the containers. The data are given below.

a. Identify the design for this experiment.
b. Write a model for this study. Identify all the terms in your model and any conditions that are placed on the terms.
c. Display estimates of all the parameters in your model.

City A Slhudge Rate 1.0 City B Sludge Rate 1.0 City C Sludge Rate 1.0 0.5 1.5 0.5 1.5 0.5 1.5 26.4 23.5 25.4 22.9 25.2 39.2 25.5 31.9 26.0 44.6 35.5 38.6 30.1 31.0 30.8 32.8 47.7 39.1 55.3 50.7 19.4 19.3 18.7 19.0 23.2 213 23.2 19.9 18.9 19.8 19.6 21.9 68.4 77.1