A scientist wants to understand the relationship between speed and position (with respect to the origin) of particles in an environment. They take a sample of 5 particles, and record their positions (x-values) and speeds (y-values):

i ,1 ,2 ,3 ,4, 5

xi ,1.21 ,0.89 ,0.06 ,0.99 .1.82

yi 2.61, 1.03, 0.02, 0.67, 3.38

( x = 0.15, y = 1.54, sx = 1.27, sy = 1.40, and rx,y = 0.22).

(a) Sketch the data and the regression line.

b) Sketch and discuss the residual plot. Remark on the appropriateness of the regression line.

The scientist transforms the data by taking x' i = x^2 i :

i, 1, 2, 3, 4, 5,

x' i, 1.46, 0.79, 0.00, 0.98, 3.31,

yi, 2.61 ,1.03, 0.02, 0.67, 3.38,

( x' = 1.31, sx' = 1.24, and rx',y = 0.92).

(c) About what percentage of the relationship between speed y and squared position x 2 is described by this new regression line?

(d) Use the new regression line to predict the speed of a particle at position 1.82.