Q.26 - The Practice of Statistics for AP

Question

Squirrels and their food supply (3.2) Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over $16$ years. Computer output from a least-squares regression on these data and a residual plot.

(a) Give the equation for the least-squares regression line. Define any variables you use. 

(b) Explain what the residual plot tells you about how well the linear model fits the data. 

(c) Interpret the values of $r^2$ and s in context. 

Step-by-Step Solution

Answer

a).  $\hat{y}=1.4146+0.4399x$ with $x$ the Cone index and $y$ the average number of offspring per female.

b). The linear model is a good model for the data.

c).  The average error made when making predictions is about $0.600309$.

Part (a) Step 1: Given Information

Given in the question that the animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over $16$ years.

Part (a) Step 2: Explanation

General least-squares equation:

$\hat{y}=a+bx$

The coefficients $a$ and $b$ are given in the column "Coef":

$a=1.4146$

$b=0.4399$

The least-squares regression equation then becomes:

$\hat{y}=1.4146+0.4399x$

with $x$ the Cone index and $y$ the average number of offspring per female.

Part (b) Step 1: Given Information

Given in the question is a computer output from a least-squares regression on these data and a residual plot. 

Part (b) Step 2: Explanation

The residual plot shows no discernible trend, and the residuals appear to be centred around $0$, indicating that the linear model is a good fit for the data.

Part (c) Step 1: Given Information

Given in the question is a computer output from a least-squares regression on these data and a residual plot. 

Part (c) Step 2: Explanation

According to the information,

$s=0.600309$

$r^2=57.2\%$

The value of $r^2=57.2\%$ means that $57.2\%$ of the variation between the variables has been explained by the linear model.

The figure of $s=0.600309$ indicates that the average forecast error is around $0.600309$.