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 The toco toucan, the largest member of the toucan 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Percent heat loss from beak 33 34 33 36 36 47 52 51 41 50 49 50 55 60 60 62
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

The toco toucan, the largest member of the toucan + x

(Use decimal notation. Give your answer to four decimal places.)

Use the equation to predict The toco toucan, the largest member of the toucan beak heat loss, as a percent of total body heat loss from all sources, at a temperature of 25 degrees Celsius.

%

What percent The toco toucan, the largest member of the toucan of the variation in beak heat loss is explained by the straight-line relationship with temperature?

%

Find the correlation The toco toucan, the largest member of the toucan (The toco toucan, the largest member of the toucan 0.001) between beak heat loss and temperature:

The population of the study is the Toco Toucan, the largest member of the toucan family. It is believed that the length and size of the beak are due to its function of dissipating heat (a cooling mechanism).

You have two variables of interest.

X: Outdoor temperature.

Temperature (ºC) 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Y: Total beak heat loss, as a percent of the total body heat loss of a toucan.

Percent heat loss from beak 33 34 33 36 36 47 52 51 41 50 49 50 55 60 60 62

The objective is to construct a least-squares regression line to predict the body heat loss of the toucans given the outdoor temperature.

Using a statistical software I estimated the regression model:

^Yi= 2.71 + 1.96Xi

1. To predict what will be the value of the beak heat loss as a percent of the total body heat loss at a temperature of 25ºC you have to simply replace the value of X in the equation:

^Yi= 2.71 + 1.96 (25)= 51.71ºC

The expected beak heat loss given an outdoor temperature of 25ºC is 51.71ºC.

2. To know what percent of the variation of the beak heat loss is explained by the outdoor temperature you have to calculate the coefficient of determination.

R²= 0.85

This means that 85% of the variability of the beak heat loss as a percent of the total body heat loss of the Toco Toucans is explained by the outdoor Temperature under the estimated model ^Yi= 2.71 + 1.96Xi

3. The correlation coefficient between these two variables is r= 0.92, this means that there is a strong positive linear correlation between the beak heat loss and the outdoor temperature. This means that when the outdoor temperature rises, the beak heat loss increases.

I hope it helps!