Hops originate from the flowers of Humulus lupulus and are used primarily as a flavoring and stability agent in beer. Hops have several characteristics that are very favorable to beer: Hops contribute a bitterness that balances the sweetness of the malt, hops can contribute aromas, and hops have an antibiotic effect that favors the activity of brewer€™s yeast over less desirable ­microorganisms. The bitterness level of a particular hop variety is measured in percent alpha acid by weight. The higher the percentage, the more bitter the hop in direct proportion. Alpha acids are now the accepted method in the brewing industry for assessing the quality of hops. The European Brewery Company carried out trials in six countries on four varieties of hops to determine if the mean temperature and mean duration of sunshine between the date of the flower coming into hop and the date of picking (the critical dates) have an impact on the alpha acid content of hops. The following data were reported by Smith in the article €œThe Influence of Temperature and Sunshine on the Alpha- Acid Content of Hops€ [Agricultural Meteorology (1974) 13: 375€“ 382]. The variables in the following table are P (alpha acid %), T (mean temperature, ° C), and S (mean sunshine, h/ day), where the means are over the critical dates. There were four varieties of hops ­included in the study.

a. Fit the model P = β0 + β1T + β2S + ɛ to the hops data with a separate equation for each variety.
b. Use residual plots, tests of hypotheses, and other diagnostic statistics to identify ­possible additional terms to add to the four models fit in part (a).

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Variety of Hops Fuggle Northerm Brewer Hallertau Field P S Field P S Field P S Field P T 12.1 16.8 4 55 6.5 4.4 68 16.7 4.4 2 5.8 174 58 2107170 622 5.3 171 58 24.9185 7.5 3 10.6 17.9 5.9 3 4.7 18.4 7.0 3 4.7 18.1 7.8 4 5.5 18.9 62410218.07.74 4.6 174 58 44.171 5.7 4.1 18.7 7.1 6 9. 21.3 6 64418.6 7.5 6 3.9 179 5.9 7 88 18.5 7.2 7 4.019.3 677 3.8 19.1 7.1 3.5 214 5.9 9 8 19.9 8.5 93221.4 6. 9 3.4 19.0 76 80 19.1 6.6 10 3.3 20.6 87 10 3.1 17.7 71 7.6 21.1 7.3 1 3.0 19.8 85 3.0 20.9 7.8 12 3.7 17.3 6.9124174 6.9 12 2.9 21.2 79 1 2.7 19.0 8.8 13 3.2 185 8613 5.8 19.2 8413 2.8 17.3 6.9 13 2.5 20 8.5 1 7.2 16.7 44 3 5.7 171 5.9 1 2 5 5.2 17.7 6.6 6 5.1 184 6.9 7 4.8 168 69 8 4.8 18.2 62 9 4.8 20.7 84 10 4.7 21.3 6210 11 4.3 21.2 7411 9.6 18.0 6.9 5 45 18.3 7.5 5 88 19.1 6,5 8 38 19.2 65 8 0