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The data at right indicate the number of injuries (y) a manufacturing company sustains when the employees work (x) concurrent tasks. As you can see, the number of injuries goes up quickly as the employees are over-worked. Tasks (x) | Injuries (y) 1 | 2 2 | 4 3 | 10 7.1 Fundamentals. The industry standard model for injuries is y = -x + 2x^2. 3 pts a) Make a (hand-drawn) to-scale x-y plot with all the following on it: - The three data points (as circles), - The industry-standard model (as a smooth curve over the range x = 0 to 4), - Three lines indicating each residual error (between each data point and the model). 1 pt b) Calculate the three residual errors between the three data points and the model. (Remember: residual errors must be left positive or negative - don't take "absolute values".) 7.2 Least-Squares Best Fit Model You're not so sure that the industry-standard model reflects your data, so you want to determine your own best-fit model. Using the general model y = px + q x^2, you want to determine the unknown coefficients p and q that give the least-squares best-fit to the three data points (xi, yi). 1 pt a) For this problem, what is the value of N? What's the value of M? 3 pts b) Write the problem in matrix form r = Y - Ac. Clearly express ... - how you are defining the vector c in terms of the two unknowns coefficients, - the values of all elements in your vector Y and matrix A from the provided data. (Don't just leave Y and A in "general" form.) 1 pt c) Develop the Normal Equations, A^T A c = A^T Y, and then solve them by hand for the two unknown coefficients. (You don't have to use Gauss Elimination - any willy-nilly elimination is fine here.) 2 pts d) Calculate the residual error vector for this model, and compare it to the residual errors you got in part (b) for 7.1. Calculate (and compare) the L2-norm of each. Which one is smaller? Does that make sense? (explain why you would expect that result) 1 pt e) Use your coefficients from (b) to write out your final "least-squares best-fit" model y(x), and apply that model to estimate the number of injuries sustained if employees worked four concurrent tasks (i.e. at x = 4).

          The data at right indicate the number of injuries (y) a manufacturing company sustains when the employees work (x) concurrent tasks. As you can see, the number of injuries goes up quickly as the employees are over-worked.

Tasks (x) | Injuries (y)
1 | 2
2 | 4
3 | 10

7.1 Fundamentals. The industry standard model for injuries is y = -x + 2x^2.
3 pts a) Make a (hand-drawn) to-scale x-y plot with all the following on it:
- The three data points (as circles),
- The industry-standard model (as a smooth curve over the range x = 0 to 4),
- Three lines indicating each residual error (between each data point and the model).
1 pt b) Calculate the three residual errors between the three data points and the model. (Remember: residual errors must be left positive or negative - don't take "absolute values".)

7.2 Least-Squares Best Fit Model
You're not so sure that the industry-standard model reflects your data, so you want to determine your own best-fit model. Using the general model y = px + q x^2, you want to determine the unknown coefficients p and q that give the least-squares best-fit to the three data points (xi, yi).
1 pt a) For this problem, what is the value of N? What's the value of M?
3 pts b) Write the problem in matrix form r = Y - Ac. Clearly express ...
- how you are defining the vector c in terms of the two unknowns coefficients,
- the values of all elements in your vector Y and matrix A from the provided data. (Don't just leave Y and A in "general" form.)
1 pt c) Develop the Normal Equations, A^T A c = A^T Y, and then solve them by hand for the two unknown coefficients. (You don't have to use Gauss Elimination - any willy-nilly elimination is fine here.)
2 pts d) Calculate the residual error vector for this model, and compare it to the residual errors you got in part (b) for 7.1. Calculate (and compare) the L2-norm of each. Which one is smaller? Does that make sense? (explain why you would expect that result)
1 pt e) Use your coefficients from (b) to write out your final "least-squares best-fit" model y(x), and apply that model to estimate the number of injuries sustained if employees worked four concurrent tasks (i.e. at x = 4).

The data at right indicate the number of injuries (y) a manufacturing company sustains when the employees work (x) concurrent tasks. As you can see, the number of injuries goes up quickly as the employees are over-worked.

Tasks (x) | Injuries (y)
1 | 2
2 | 4
3 | 10

7.1 Fundamentals. The industry standard model for injuries is y = -x + 2x^2.
3 pts a) Make a (hand-drawn) to-scale x-y plot with all the following on it:
- The three data points (as circles),
- The industry-standard model (as a smooth curve over the range x = 0 to 4),
- Three lines indicating each residual error (between each data point and the model).
1 pt b) Calculate the three residual errors between the three data points and the model. (Remember: residual errors must be left positive or negative - don't take "absolute values".)

7.2 Least-Squares Best Fit Model
You're not so sure that the industry-standard model reflects your data, so you want to determine your own best-fit model. Using the general model y = px + q x^2, you want to determine the unknown coefficients p and q that give the least-squares best-fit to the three data points (xi, yi).
1 pt a) For this problem, what is the value of N? What's the value of M?
3 pts b) Write the problem in matrix form r = Y - Ac. Clearly express ...
- how you are defining the vector c in terms of the two unknowns coefficients,
- the values of all elements in your vector Y and matrix A from the provided data. (Don't just leave Y and A in "general" form.)
1 pt c) Develop the Normal Equations, A^T A c = A^T Y, and then solve them by hand for the two unknown coefficients. (You don't have to use Gauss Elimination - any willy-nilly elimination is fine here.)
2 pts d) Calculate the residual error vector for this model, and compare it to the residual errors you got in part (b) for 7.1. Calculate (and compare) the L2-norm of each. Which one is smaller? Does that make sense? (explain why you would expect that result)
1 pt e) Use your coefficients from (b) to write out your final "least-squares best-fit" model y(x), and apply that model to estimate the number of injuries sustained if employees worked four concurrent tasks (i.e. at x = 4).

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