Question

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 from all sources, at various temperatures in degrees Celsius. [Note: The numerical values in this problem have been modified for testing purposes.]

Temperature (oC)(oC)	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Percent heat loss from beak	32	34	34	34	35	48	55	50	42	50	43	54	57	58	63	63

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: (Use decimal notation. Enter the values of the intercept and slope rounded to two decimal places. Use the letter ?x to represent the value of the temperature.)

?̂ =y^=

Use the equation of the least‑squares regression line to predict beak heat loss, as a percent of total body heat loss from all sources, at a temperature of 2525 degrees Celsius. Enter your answer rounded to two decimal places.

beak heat as a percent of total body heat loss=beak heat as a percent of total body heat loss=

%

What percent of the variation in beak heat loss is explained by the straight-line relationship with temperature? Enter your answer rounded to two decimal places.

percent of variation in beak heat loss explained by the equation=percent of variation in beak heat loss explained by the equation=

%

Find the correlation ?r between beak heat loss and temperature. Enter your answer rounded to three decimal places.

?=

Answer 1

Solution:

Regression equation:
y = 0.743 + 2.056⋅x
when x=25
y = 0.743 + 2.056⋅x
= 0.743 + 2.056(25) = 52.143

Coefficient of determination, R^2 = 0.8064 or 80.64%

Correlation coefficient, r = sqrt(0.8064) = 0.898