Least Squares and Applications

The best answer when there is no exact one.

12 problems · about 24 minutes · S-ID.B.6, N-VM.C.11

NameDate

Figure — use these to answer the problems

No line passes through all seven points, so no exact solution exists. Move the line until the gaps look as small as you can make them.

y = x + 1

slope1

intercept1

Figure 1

Straightforward practice. Get the method working first.

5 problems

Residuals of 6 and 8. What is the total of their squares?
Answer
2
What does least squares find?
- The x making Ax as close to b as possible, when no exact solution exists.
- An exact solution that ordinary methods missed.
A prediction of 20 against an observed 18. What is the residual?
Answer
Residuals of 3 and 4. What is the total of their squares?
Answer
Least squares minimises the total of the squared what? Enter 1 for residuals, 2 for slopes.
Answer

The same ideas with more to keep track of.

3 problems

Every residual is 0. Was the system solvable exactly? 1 yes, 0 no.
Answer
The closest reachable vector to b is its projection onto what? Enter 1 for the column space, 2 for the null space.
Answer
Residuals of 5 and 12. What is the total of their squares?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order the reasoning behind least squares.
Write 1 to 4 in the boxes to put these in order.
- Notice that Ax = b has no exact solution
- Ask instead for the closest reachable Ax
- Project b onto the column space of A
- Solve for the x that produces that projection
Is a line of best fit a least squares solution? 1 yes, 0 no.
Answer
The Overdetermined System
40 equations in 3 unknowns. Is an exact solution likely? 1 yes, 0 no.
Answer
The Residual
A prediction of 12 against an observed value of 15. What is the residual?
Answer