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 70 inches tall and the residual for his son’s height is 2.5. what is the son’s actual height? \\(\bigcirc\\) 71.7 inches \\(\bigcirc\\) 74.2 inches \\(\bigcirc\\) 76.7 inches \\(\bigcirc\\) 82.3 inches

Explanation:

Step1: Recall the residual formula

The residual \( e \) is given by the formula \( e = y - \hat{y} \), where \( y \) is the actual value, and \( \hat{y} \) is the predicted value. We can rearrange this formula to solve for \( y \): \( y = \hat{y} + e \).

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

We know the regression line is \( \hat{y} = -2.8 + 1.1x \), and the father's height \( x = 70 \) inches. Substitute \( x \) into the regression line:
\[
\hat{y} = -2.8 + 1.1 \times 70
\]
First, calculate \( 1.1 \times 70 = 77 \), then \( \hat{y} = -2.8 + 77 = 74.2 \) inches.

Step3: Calculate the actual height (\(y\))

We know the residual \( e = 2.5 \). Using the formula \( y = \hat{y} + e \), substitute \( \hat{y} = 74.2 \) and \( e = 2.5 \):
\[
y = 74.2 + 2.5 = 76.7
\]

Answer:

76.7 inches (corresponding to the option "76.7 inches")