Question

the students in a precalculus class measured each students height and arm span, in centimeters. the students calculated a linear regression ( y = a + bx ) with heights as the input values and arm spans as the output values. the given residual plot has a point labeled ( p ) at coordinates ( (175,23.4) ). what does point ( p ) indicate in the context?
a because point ( p ) is above the ( x )-axis, for the student with a height of ( 175 mathrm{~cm} ), the model overestimates the actual arm span value by ( 23.4 mathrm{~cm} ).
b because point ( p ) is above the ( x )-axis, for the student with a height of ( 175 mathrm{~cm} ), the model underestimates the actual arm span value by ( 23.4 mathrm{~cm} ).
c because point ( p ) is above the ( x )-axis, for the student with an arm span of ( 175 mathrm{~cm} ), the model overestimates the actual height value by ( 23.4 mathrm{~cm} ).
d because point ( p ) is above the ( x )-axis, for the student with an arm span of ( 175 mathrm{~cm} ), the model underestimates the actual height value by ( 23.4 mathrm{~cm} ).

Explanation:

In a residual plot, the \(x\) - value represents the independent variable (here, height) and the \(y\) - value represents the residual. The formula for the residual is \(e=y - \hat{y}\), where \(y\) is the actual value and \(\hat{y}\) is the predicted value. If the residual \(e>0\) (point above the \(x\) - axis), then \(y-\hat{y}>0\) or \(y>\hat{y}\), which means the model underestimates the actual value. Here, the independent variable (input) is height (\(x = 175\) cm) and the dependent variable (output) is arm - span.

Answer:

B. Because point \(P\) is above the \(x\) - axis, for the student with a height of \(175\) cm, the model underestimates the actual arm span value by \(23.4\) cm.