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, at various temperatures in degrees Celsius:
Temperature 15 16 17 18 19 20 21 22 23 24 25 26 27
Percent heat loss from beak 33 34 33 36 36 47 52 51 41 50 49 50 55
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 ______.

Answer 1

Answer:

$m=\frac{332}{182}=1.824$

$b=\bar y -m \bar x=43.615-(1.824*21)=5.311$

So the line would be given by:

$y=1.824 x +5.311$

Step-by-step explanation:

We assume that the data is this one:

x: 15 16 17 18 19 20 21 22 23 24 25 26 27

y: 33 34 33 36 36 47 52 51 41 50 49 50 55

Find the least-squares line appropriate for this data.  

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i =273$

$\sum_{i=1}^n y_i =567$

$\sum_{i=1}^n x^2_i =5915$

$\sum_{i=1}^n y^2_i =25547$

$\sum_{i=1}^n x_i y_i =12239$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=5915-\frac{273^2}{13}=182$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=12239-\frac{273*567}{13}=332$

And the slope would be:

$m=\frac{332}{182}=1.824$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{273}{13}=21$

$\bar y= \frac{\sum y_i}{n}=\frac{567}{13}=43.615$

And we can find the intercept using this:

$b=\bar y -m \bar x=43.615-(1.824*21)=5.311$

So the line would be given by:

$y=1.824 x +5.311$