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) 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Percent heat loss from beak 33 32 33 32 39 43 56 48 44 49 44 56 58 60 60 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 ? to represent the value of the temperature.) ?̂ = 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 25 degrees Celsius. Enter your answer rounded to two decimal places. 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= % Find the correlation ? between beak heat loss and temperature. Enter your answer rounded to three decimal places. ?=

Answer 1

from the given data of information

X	Y	XY	X2	Y2
15	33	495	225	1089
16	32	512	256	1024
17	33	561	289	1089
18	32	576	324	1024
19	39	741	361	1521
20	43	860	400	1849
21	56	1176	441	3136
22	48	1056	400	2304
23	44	1012	529	1936
24	49	1176	576	2401
25	44	1100	625	1936
26	56	1456	676	3136
27	58	1566	784	3364
28	60	1680	841	3600
29	60	1740	841	3600
30	63	1890	900	3969

BY using the above formula we get

n	12
sum(xy)	11776.42
sum(x)	273.00
sum(y)	525.00
sum(x2)	5926.00
sum(y2)	26193.00

b = 0.5876

a = 3.3821

the equation of regression line y=a+bx

by substituting the above a and b values in the regression line we get

y = 3.3821+ 0.5876 x

the percent of total body heat loss from all sources when x is 25

y=18

43.97% of the variation in break heat is explained by the straight line relationship with temperature

correlation r is approximately 0.1746