b Here are the values of x log x and y log y x 070 100 118 130 140 148 165 178

B here are the values of x log x and y log y x 070

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b. Here are the values of x log( x ) and y log( y ): x 0.70 1.00 1.18 1.30 1.40 1.48 1.65 1.78 y 1.21 0.99 0.91 0.62 0.53 0.46 0.28 0.11 Construct a scatterplot of these transformed data, and comment on the pattern. c. Based on the accompanying MINITAB output, does the least-squares line effectively summarize the relationship between y and x ? The regression equation is log(moisture) = 2.02 – 1.05 log(time) Predictor Coef SE Coef T P Constant 2.01780 0.09584 21.05 0.000 log(time) –1.05171 0.07091 –14.83 0.000 S = 0.0657067 R-Sq = 97.3% R-Sq(adj) = 96.9% Analysis of Variance Source DF SS MS F P Regression 1 0.94978 0.94978 219.99 0.000 Residual Error 6 0.02590 0.00432 Total 7 0.97569 d. Use the MINITAB output to predict moisture content when frying time is 35 sec. e. Do you think that predictions of moisture content using the model in Part (c) will be better than those using the model fit in Example 5.16, which used transformed y val- ues but did not transform x ? Explain. 5.53 The report “Older Driver Involvement in Injury Crashes in Texas” (Texas Transportation Institute, 2004) included a scatterplot of y fatality rate (percentage of drivers killed in injury crashes) versus x driver age. The accompanying data are approximate values read from the scatterplot. Fatality Fatality Age Rate Age Rate 40 0.75 80 2.20 45 0.75 85 3.00 50 0.95 90 3.20 55 1.05 a. Construct a scatterplot of these data. b. Using Table 5.5 and the ladder of transformations in Figure 5.31, suggest a transformation that might result in variables for which the scatterplot would exhibit a pattern that was more nearly linear. c. Reexpress x and/or y using the transformation you rec- ommended in Part (b). Construct a scatterplot of the trans- formed data. 252 C h a p t e r 5 Summarizing Bivariate Data The corresponding least-squares line is This transformation can be reversed by squaring both sides to obtain an equation of the form y some function of x : Since y 2 y and we get y ˆ 1 2.45 3.72 1 x 2 2 x ¿ 1 x y ˆ ¿ 2 1 2.45 3.72 x ¿ 2 2 y ˆ ¿ 2.45 3.72 x ¿ .............................................................................................................. Bold exercises answered in back Data set available online but not required Video solution available
d. Does the scatterplot in Part (c) suggest that the trans- formation was successful in straightening the plot ? e. Using the transformed variables, fit the least-squares line and use it to predict the fatality rate for 78-year-old drivers. 5.54 The paper “Aspects of Food Finding by Wintering Bald Eagles” ( The Auk [1983]: 477–484) examined the relationship between the time that eagles spend aerially searching for food (indicated by the percentage of eagles soaring) and relative food availability. The accompanying data were taken from a scatterplot that appeared in this pa- per. Let x denote salmon availability and y denote the per- centage of eagles in the air. x 0 0 0.2 0.5 0.5 1.0 y 28.2 69.0 27.0 38.5 48.4 31.1 x 1.2 1.9 2.6 3.3 4.7 6.5 y 26.9 8.2 4.6 7.4 7.0 6.8 a. Draw a scatterplot for this data set. Would you describe the plot as linear or curved ? b. One possible transformation that might lead to a straighter plot involves taking the square root of both the x and y values. Use Figure 5.31 to explain why this might be a reasonable transformation.

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