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	34	35	33	33	35	48	54	52	45	50	44	54	59	62	64	64

1) 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^=

2) 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=

3) 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=

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

r=

Answer 1

( 1 ) Equation of the least-squares regression line:

Find X⋅Y and X2 as it was done in the table below.

X	Y	X⋅Y	X⋅X
15	34	510	225
16	35	560	256
17	33	561	289
18	33	594	324
19	35	665	361
20	48	960	400
21	54	1134	441
22	52	1144	484
23	45	1035	529
24	50	1200	576
25	44	1100	625
26	54	1404	676
27	59	1593	729
28	62	1736	784
29	64	1856	841
30	64	1920	900

Y = -0.90 + 2.17 X

( 2 ) Given x = 25

then

y = -0.90 + 2.17 ( 25 )

= 53.35

y = 53.35

( 3 )

r = 0.9099

r² = 82.79 %

4) from part ( 3 ),

r = 0.910