Math 205B&C 04/03/09 Quiz 08 page 1 Name 8am 1:10 pm
1. Consider the points P1=(1,−16), P2=(3,10), P3=(7,8), and P4=(9,10), measurements made in
a lab where the data was supposed lie on a single line. Obviously something went wrong; these four points
can’t possibly be on the same line.
1a. Explicitly, what are the design matrix Xand observation vector yyou would use in a matrix
equation Xβ=yto find β=β0
β1such that the four points above are on the line y=β0+β1x?
X=y=
1b. Since Xβ=yhas no solution, we’ll have to be content with the least-squares solution. Find it; ie,
find β0and β1for which y=β0+β1xis the least-squares line that best fits the four given points. Show
all your work!
β0=β1=
1c. Find the four ycoordinates corresponding to x=1,3,7,9 on the best-fit line, and assemble them
into a vector we’ll call p;pis our vector of predicted values.
p=
1d. Find the four residuals, and then the sum of their squares (SOS).
Four Residuals: SOS:
1e. You might think that the line y=−77/4+13/4xwould be a better fit: At least it goes through
P1and P4. Find the four residuals for this line, and then the sum of their squares.
Four Residuals: SOS:
1f. Which line is a better fit? Explain.