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. Presented is the 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 (°C)	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

(a) What is 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? (Use decimal notation. Enter the values of the intercept and slope rounded to two decimal places. Enter "x

" to represent the value of the temperature.)

^y=

What does the slope of the least‑squares line say about the relationship between beak heat loss and temperature? Select the correct explanation.

For every degree Celsius, the toucan will lose about 2.06%

more heat through its beak.

For every degree Celsius, the toucan will lose about 0.74%

more heat through its beak.

None of the options are correct.

For every degree Celsius, the toucan will lose about 18.5%

more heat through its beak.

(b) 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 °C.

(Enter your answer rounded to two decimal places.)

beak heat as a percent of total body heat loss=

%

(c) 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=

%

(d) Find the correlation r

between beak heat loss and temperature. (Enter your answer rounded to three decimal places.)

r=

Answer 1

a.

Sum of X = 360
Sum of Y = 752
Mean X = 22.5
Mean Y = 47
Sum of squares (SS_X) = 340
Sum of products (SP) = 699

Regression Equation = ŷ = bX + a

b = SP/SS_X = 699/340 = 2.06

a = M_Y - bM_X = 47 - (2.06*22.5) = 0.74

ŷ = 2.06X + 0.74

Here we see that slope is 2.06

Hence answer is

For every degree Celsius, the toucan will lose about 2.06% more heat through its beak.

b. For x=25, ŷ = (2.06*25) + 0.74=52.24

beak heat as a percent of total body heat loss= 52.24%

c. Here first we will find r and as we know percentage of variation in y explained by x is given by r^2

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

Y Values
∑ = 752
Mean = 47
∑(Y - M_y)² = SS_y = 1782

X and Y Combined
N = 16
∑(X - M_x)(Y - M_y) = 699

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

r = 699 / √((340)(1782)) = 0.8980

So r^2=0.8980^2=0.8064

percent of variation in beak heat loss explained by the equation=80.64%

d. As computed in c, we get r=0.898