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 ý = -
2.8+1.1x, where ŷ 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?
O
71.7 inches
O
74.2 inches
O
76.7 inches
O 82.3 inches

Answer 1

The son's actual height is found by adding the residual value to the son's predicted height obtained from the least-squares regression line. The son's actual height is 76.7 inches when the father is 70 inches tall and the residual is 2.5 inches.

Step-by-step explanation:

The student asked how to find the actual height of a son given the father's height and the residual in the context of a least-squares regression line. The regression line is ý = -2.8 + 1.1x, where ý represents the predicted height of the son (in inches) based on the father's height x (also in inches). The father is 70 inches tall, and the residual for his son's height is 2.5 inches. To find the son's actual height, we first use the regression equation to calculate the predicted height of the son:

Predicted height (ý) = -2.8 + 1.1(70) = -2.8 + 77 = 74.2 inches.

Then, we add the residual to the predicted height to obtain the actual height:

Actual height = Predicted height + Residual = 74.2 inches + 2.5 inches = 76.7 inches.

Therefore, the son's actual height is 76.7 inches.

Answer 2

the son is 76.5 inches

Step-by-step explanation:

We know that the predicted height of the son based on the father's height is given by the equation ŷ = -2.8 + 1.1x, where x is the height of the father. We are also given that the residual for the son's height of one father who is 70 inches tall is 2.5.

The residual is the difference between the actual value of y and the predicted value of y. So we can set up the equation:

y - ŷ = residual

where y is the actual height of the son.

Substituting the given values, we get:

y - (-2.8 + 1.1(70)) = 2.5

Simplifying, we get:

y - 74 = 2.5

y = 76.5

Therefore, the actual height of the son is 76.5 inches.