Question

The toco toucan, Ramphastos toco, is the largest member of the toucan family and possesses the largest beak relative to body size of all birds. This exaggerated feature has been interpreted in various ways, 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.
The table provided contains data for beak heat loss, as a percent of total body heat loss, at various temperatures in degrees Celsius:

Temperature (∘C) 1515 1616 1717 1818 1919 2020 2121 2222
Percent heat loss from beak 3232 3434 3535 3333 3737 4646 5555 5151
Temperature (∘C) 2323 2424 2525 2626 2727 2828 2929 3030
Percent heat loss from beak 4343 5252 4545 5353 5858 6060 6262 6262

Required:
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 temperature?
b. What percent of the variation in beak heat loss is explained by the straight‑line relationship with temperature?

Answer 1

Answer:

(a) $\hat y=0.919+2.065x$

(b) 83.6%

Step-by-step explanation:

(a)

A least square regression line is to be formed for predicting beak heat loss, as a percent of total body heat loss, from temperature.

Use MS-Excel to form the regression line.

Go to Data → Data Analysis → Regression

A dialog box will open.

Select the X and Y variables.

Press OK.

The regression output is attached below.

The equation of the least‑squares regression line is:

$\hat y=0.919+2.065x$

(b)

The coefficient of determination R² specifies the percentage of the variance in the dependent variable (Y) that is forecasted or explained by linear regression and the forecaster variable (X, also recognized as the independent variable).

Consider the regression output attached below.

The R² value is 0.8360.

That is 83.6% of the variation in beak heat loss is explained by the straight‑line relationship with temperature.