The following data set has ratings from ratebeer.com along with values of IBU (international bittering units, a measure of bitterness) and ABV (alcohol by volume) for 25 beers. Notice which beers have the lowest ratings and which are highest.

a. Find the correlations (and the corresponding P-values) among Rating, IBU, and ABV.
b. Regress rating on IBU and ABV. Notice that although both predictors have strongly significant correlations with Rating, they do not both have significant regression coefficients. How do you explain this?
c. Plot the residuals from the regression of (b) to check the assumptions. Also plot rating against each of the two predictors. Which of the assumptions is clearly not satisfied?
d. Regress rating on IBU and ABV with the square of IBU as a third predictor. Again check assumptions.
e. How effective is the regression in (d)? Interpret the coefficients with regard to statistical significance and sign. In particular, discuss the relationship to IBU.
f. Summarize your conclusions.

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Beer Amstel Light Anchor Liberty Ale Anchor Steam Bud Light Budweiser Coors DAB Dark Dogfish 60 min IPA Great Divide Titan IPA Great Divide Hercules Double IPA Guinness Extra Stout Harp Lager Heineken Heineken Premium Light Michelob Ultra Newcastle Brown Ale Pilsner Urquell Redhook ESB Rogue Imperial Stout Samuel Adams Boston Lager Shiner Light Sierra Nevada Pale Ale Sierra Nevada Porter Terrapin All-Amer. Imperial Pilsner Three Floyds Alpha King IBU ABV Rating 18 1.93 54 3.60 3.31 1.15 1.38 1.63 2.73 3.76 33 7 11 14 32 60 65 85 60 21 23 11 4 18 35 29 88 31 13 37 40 75 66 3.5 5.9 4.9 4.2 5 5 5 6 6.8 9.1 5 4.3 5 3.2 4.2 4.7 4.4 5.77 11.6 4.9 4.03 5.6 5.6 7.5 6 3.81 4.05 3.38 2.85 2.13 1.62 1.01 3.05 3.28 3.06 3.98 3.19 2.57 3.61 3.60 3.46 4.04