Chapter 28, Problem 28.1AYK

To determine

To write a population regression model that describe this setting and explain in words what each $\beta$ in your model means.

Explanation of Solution

In the question, it is given that suppose that the number y of new birds that join a colony this year has the same straight line relationship with the percent $x_1$ of returning birds in colonies for two different bird species. An indicator variable shows which species we observe; $x_2=0$ for one and $x_2=1$ for the other. Let us consider the population regression model be as:

  ${\mu }_y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$

Where the number y is new birds that join a colony this year, $x_1$ isthe percent of returning birds in colonies and the indicator $x_2$ is the different species of bird returning with $x_2=0$ for one and $x_2=1$ for the other. So, for the one species with $x_2=0$ for one, the model becomes,

  $\begin{array}{l}{\mu }_y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2\\ ={\beta }_0+{\beta }_1{x}_1+{\beta }_2(0)\\ ={\beta }_0+{\beta }_1{x}_1\end{array}$

And for the other species with $x_2=1$ for the other, the model becomes:

  $\begin{array}{l}{\mu }_y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2\\ ={\beta }_0+{\beta }_1{x}_1+{\beta }_2(1)\\ ={\beta }_0+{\beta }_1{x}_1+{\beta }_2\\ =({\beta }_0+{\beta }_2)+{\beta }_1{x}_1\end{array}$

And in these we can see that the slope that describes how the number of new birds that join a colony this year as percent $x_1$ varies is ${\beta }_1$ . The intercept differ: ${\beta }_0$ for the one species and $({\beta }_0+{\beta }_2)$ for the other species. So, ${\beta }_2$ is of particular interest because it is the fixed change between one and the other species of new birds that join a colony this year.