Question

vera rents bicycles to tourists. she recorded the height (in cm) of each customer and the frame size (in 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: $hat{y}=\frac{1}{3}+\frac{1}{3}x$. what is the residual of a customer with a height of 155 cm who rents a bike with a 51 cm frame?

Explanation:

Step1: Predict frame size

Substitute $x = 155$ into the regression equation $\hat{y}=\frac{1}{3}+\frac{1}{3}x$.
$\hat{y}=\frac{1}{3}+\frac{1}{3}\times155=\frac{1 + 155}{3}=\frac{156}{3}=52$.

Step2: Calculate residual

The formula for residual is $e=y-\hat{y}$, where $y = 51$ (actual frame - size) and $\hat{y}=52$ (predicted frame - size).
$e=51 - 52=-1$.

Answer:

$-1$