Question

Vera rents bicycles to tourists. She recorded the height (in \[\text{cm}\]) of each customer and the frame size (in \[\text{cm}\]) of the bicycle that customer rented. After plotting her results, Vera noticed that the relationship between the two variables was fairly linear, so she used the data to calculate the following least squares regression equation for predicting bicycle frame size from the height of the customer:

Y= 1/3 + 1/3x
What is the residual of a customer with a height of 155 cm who rents a bike with a 51 cm frame?

Answer 1

The residual for a customer with a height of 155 cm who rents a bicycle with a 51 cm frame size is -1 cm. This was calculated using Vera's least-squares regression equation, and the negative value indicates that the actual frame size is less than predicted.

Step-by-step explanation:

Vera has calculated a least-squares regression equation to predict bicycle frame size from a customer's height, which is Y = \( \frac{1}{3} \) + \( \frac{1}{3}x \). To find the residual for a customer with a height of 155 cm, we first use the equation to predict the frame size for this height. The predicted frame size is:

\( \hat{Y} = \frac{1}{3} + \frac{1}{3}(155) = \frac{1}{3} + \frac{155}{3} \)

\( \hat{Y} = \frac{1 + 155}{3} = \frac{156}{3} = 52 \) cm

The actual frame size rented was 51 cm. Thus, the residual is the actual frame size minus the predicted frame size:

Residual = Actual size - Predicted size

Residual = 51 cm - 52 cm = -1 cm

The negative sign indicates that the actual frame size is 1 cm less than what was predicted by the regression equation.