Question

5.33) The toco toucan, the largest member of the toucan family, possesses the largest beak relative to body size of all birds. This exaggerated feature has received various interpretations, such as being a refined adaptation for feeding. However, the large surface area may also be an important mechanism for radiating heat (and hence cooling the bird) as outdoor temperature increases. Here are data for beak heat loss, as a percent of total body heat loss, at various temperatures in degrees Celsius: Temperature(oC) ( o C ) 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Percent heat loss from beak 36 38 31 37 39 48 52 47 39 53 47 49 53 58 60 61 The equation of the least-squares regression line for predicting beak heat loss, as a percent of total body heat loss from all sources, from temperature is yˆ= y ^ = + x (Use decimal notation. Give your answer to four decimal places.) Use the equation to predict (±0.01) ( ± 0.01 ) beak heat loss, as a percent of total body heat loss from all sources, at a temperature of 25 degrees Celsius. % What percent (±0.01) ( ± 0.01 ) of the variation in beak heat loss is explained by the straight-line relationship with temperature? % Find the correlation r r (± ± 0.001) between beak heat loss and temperature: r= r =

Answer 1

Sum of X = 360
Sum of Y = 748
Mean X = 22.5
Mean Y = 46.75
Sum of squares (SS_X) = 340
Sum of products (SP) = 581

Regression Equation = ŷ = bX + a

b = SP/SS_X = 581/340 = 1.7088

a = M_Y - bM_X = 46.75 - (1.71*22.5) = 8.3015

ŷ = 1.7088X + 8.3015

For x=25, ŷ = (1.7088*25) + 8.3015=51.0215=51

X Values
∑ = 360
Mean = 22.5
∑(X - M_x)² = SS_x = 340

Y Values
∑ = 748
Mean = 46.75
∑(Y - M_y)² = SS_y = 1273

X and Y Combined
N = 16
∑(X - M_x)(Y - M_y) = 581

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 581 / √((340)(1273)) = 0.8831

So r^2=0.8831^2=0.7799

So 77.99% of variation in beak heat loss is explained by the straight-line relationship with temperature.

As we have calculated r to find the r^2, so answer for r=0.8831