Can you tell how old a lion is by looking at its nose? A professor at the University of Wisconsin-Madison conducted a study of data taken from 32 lions and observed the relationship between age (in years) and proportion of blackness in the lion’s nose. The equation of the least squares regression line was y hat = 0.8790 + 10.6471x where y n is the predicted age of the lion, measured in years, and x is the proportion of the lion’s nose that is black. A lion whose nose was 11% black was known to be 1.9 years old. What is the residual for the age of this lion?

In regression, the difference between the observed value of the dependent variable (y) and the predicted value ([tex]\hat y[/tex]) is known as the residual (e).

[tex]e=y-\hat y[/tex]

The least square regression line is used to predict the value of the response or dependent variable (y) from the known value of the explanatory or independent variable (x).

The general form of a least square regression line is:

[tex]\hat y=\alpha +\beta x[/tex]

The equation of the least squares regression line to predict the relationship between age (in years) and proportion of blackness in the lion’s nose is:

[tex]\hat y=0.8790+10.6471 x[/tex]

Compute the predicted value of y for x = 0.11 as follows:

[tex]\hat y=0.8790+10.6471 x[/tex]

[tex]=0.8790+(10.6471\times 0.11)\\=0.8790+1.171181\\=2.050181\\\approx 2.05[/tex]

The predicted value of y is, [tex]\hat y=2.05[/tex].

The observed value of the age of lion whose nose is 11% black is, y = 1.90.

Compute the residual age of this lion as follows:

[tex]e=y-\hat y[/tex]

[tex]=1.90-2.05\\=-0.15[/tex]

Thus, the residual age of a lion whose nose is 11% black and is 1.9 years old is -0.15.