Question

an anthropologist is interested in the relationship between fathers and sons heights. she collects a simple random sample of 25 fathers and 25 sons, and determines that the least-squares regression line is \\(\hat{y} = -2.8 + 1.1x\\), where \\(\hat{y}\\) is the predicted height of each son and \\(x\\) is the height of his father (both measured in inches). one father is 72 inches tall, and his son is 75 inches tall. what is the residual for the sons height? \\(\bigcirc -2.8\\) \\(\bigcirc -1.4\\) \\(\bigcirc 1.1\\) \\(\bigcirc 1.4\\)

Explanation:

Step1: Recall the residual formula

The residual is calculated as the actual value minus the predicted value, i.e., \( \text{Residual} = y - \hat{y} \), where \( y \) is the actual height of the son and \( \hat{y} \) is the predicted height from the regression line.

Step2: Find the predicted height (\(\hat{y}\))

Given the regression line \( \hat{y} = -2.8 + 1.1x \), and the father's height \( x = 72 \) inches. Substitute \( x = 72 \) into the regression line:
\[
\hat{y} = -2.8 + 1.1 \times 72
\]
First, calculate \( 1.1 \times 72 = 79.2 \), then:
\[
\hat{y} = -2.8 + 79.2 = 76.4
\]

Step3: Calculate the residual

The actual height of the son \( y = 75 \) inches. Using the residual formula:
\[
\text{Residual} = y - \hat{y} = 75 - 76.4 = -1.4
\]

Answer:

-1.4