An article in Biotechnology Progress (December 2002, Vol. 18(6), pp. 1170€“1175) presented a 2^7€“3fractional factorial to evaluate factors promoting astaxanthin production. The data are shown in the following table.

The factors and levels are shown in the following table.

(a) Write down the complete defining relation and the aliases from the design.

(b) Estimate the main effects.

(c) Plot the effect estimates on normal probability paper and interpret the results.

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A B C D E F G Weight Content (mg/g) Runs Cellular Content (pg/cell) 4.2 10.8 -1 -1 -1 -1 1 1 4.4 24.9 1. 27.3 -1 1. -1 -1 1 7.8 4 1 -1 1 -1 14.9 36.3 -1 -1 1 -1 25.3 112.6 1 -1 1 -1 1 26.7 159.3 145.2 -1 1 1 -1 23.9 -1 243.2 1 1 1 21.9 1 1 -1 24.3 72.1 10 -1 1 1 1 1 20.5 112.2 22.5 11 1 -1 1 1 -1 10.8 -1 12 20.8 149.7 13 -1 1 -1 13.5 140.1 14 -1 -1 1 -1 1 1 10.3 47.3 23.0 15 -1 -1 -1 153.2 16 -1 -1 -1 -1 -1 -1 -1 12.1 35.2