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Two researchers studied the relationship between body weight and brain weight of 62 species of land mammals

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Two researchers studied the relationship between body weight and brain weight of 62 species of land mammals. The response Y and the predictor X are two indices representing the brain weight and the body weight, respectively. (In case you are wondering, they are logarithms of the original measurements, hence the negative numbers.) Larger values of an index indicate heavier weights. The figure below is a scatter plot superimposed with a regression line. A summary of the linear regression from R is given below.

Coefficients:      

Estimate Std.     Error                   t value   Pr(>|t|)

(Intercept) 0.92713         0.04171             22.23     <2e-16 ***

body     0.75169               0.02846             26.41     <2e-16 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3015 on 60 degrees of freedom

Multiple R-squared: 0.9208, Adjusted R-squared: 0.9195

F-statistic: 697.4 on 1 and 60 DF, p-value: < 2.2e-16

(a) Let yb be the fitted brain weight index, x be the body weight index. Based on the output, establish the regression line in the form of yb = b0 + b1x.

(b) Does b0 have a meaningful interpretation in this context? If so, write thorough interpretation. If not, why not?

(c) Write thorough interpretation of b1.

(d) Human has a body weight index of 1.79. What is the predicted brain weight index yb for human based on the regression line?

(e) In fact, the actual brain weight index for human is y = 3.12. What is the value of the residual (prediction error) y − yb?

(f) What is the value of the coefficient of determination r 2 ?

(g) Based on the value of r 2 , what is the correlation coefficient r between brain and body weight indices?

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