Problem 1 [40 points]. The horsepower developed by an automobile engine on a dynamometer is thought to be a function of the engine speed in revolutions per minute (rpm), the road octane number of the fuel, and the engine compression. An experiment is run in the laboratory and the following data are collected.
| Factor 1 | Factor 2 | Factor 3 | Response |
| Engine speed [rpm] | Fuel type [octane number] | Engine compression [psi] | Dynamometer [horsepower] |
| 2000 | 90 | 100 | 225 |
| 1800 | 94 | 95 | 212 |
| 2400 | 88 | 110 | 229 |
| 1900 | 91 | 96 | 222 |
| 1600 | 86 | 100 | 219 |
| 2500 | 96 | 110 | 278 |
| 3000 | 94 | 98 | 246 |
| 3200 | 90 | 100 | 237 |
| 2800 | 88 | 105 | 233 |
| 3400 | 86 | 97 | 224 |
| 1800 | 90 | 100 | 223 |
| 2500 | 89 | 104 | 230 |
- [10pt] Fit a multiple linear regression model to the data.
- [10pt] Test the regression significance
- [10pt] Evaluate the residuals and comment on model accuracy
- [10pt] Redo the analysis by adding interaction terms to the model
Problem 2 [20 points] Consider the three-variable central composite design shown in the following table with 3 factors and two responses – conversion % and quantity. Analyze the data and draw appropriate conclusions, assuming that we wish to maximize conversion (y1) with quantity (y2) between 55 and 60 .
| Run | Time [min] | Temperature [oC] | Catalyst [%] | Conversion [%] | Activity [qty] |
| A | B | C | y1 | y2 | |
| 1 | -1 | -1 | -1 | 74 | 53.2 |
| 2 | 1 | -1 | -1 | 51 | 62.9 |
| 3 | -1 | 1 | -1 | 88 | 53.4 |
| 4 | 1 | 1 | -1 | 70 | 62.6 |
| 5 | -1 | -1 | 1 | 71 | 57.3 |
| 6 | 1 | -1 | 1 | 90 | 67.9 |
| 7 | -1 | 1 | 1 | 66 | 59.8 |
| 8 | 1 | 1 | 1 | 97 | 67.8 |
| 9 | 0 | 0 | 0 | 81 | 59.2 |
| 10 | 0 | 0 | 0 | 75 | 60.4 |
| 11 | 0 | 0 | 0 | 76 | 59.1 |
| 12 | 0 | 0 | 0 | 83 | 60.6 |
| 13 | -1.682 | 0 | 0 | 76 | 59.1 |
| 14 | 1.682 | 0 | 0 | 79 | 65.9 |
| 15 | 0 | -1.682 | 0 | 85 | 60 |
| 16 | 0 | 1.682 | 0 | 97 | 60.7 |
| 17 | 0 | 0 | -1.682 | 55 | 57.4 |
| 18 | 0 | 0 | 1.682 | 81 | 63.2 |
| 19 | 0 | 0 | 0 | 80 | 60.8 |
| 20 | 0 | 0 | 0 | 91 | 58.9 |
- [5pts] Develop quadratic models for Conversion and Quantity including quadratic and two factor interactions.
- [5pts] Reduce the quadratic model by eliminating insignificant terms as appropriate
- [5pts] Sketch a contour plot for Conversion and Quantity respectively
- [5pts] Sketch the overlay plot
Problem 3 [30 points] A chemical engineer collected the following empirical process operation data. The response y is filtration time, while coded variable x1 is temperature, and x2 is pressure.
| A: Temperature | B: Pressure | R1: Filtration Time |
| x1 | x2 | y |
| -1 | -1 | 54 |
| -1 | 1 | 45 |
| 1 | -1 | 32 |
| 1 | 1 | 47 |
| -1.414 | 0 | 50 |
| 1.414 | 0 | 53 |
| 0 | -1.414 | 47 |
| 0 | 1.414 | 51 |
| 0 | 0 | 41 |
| 0 | 0 | 39 |
| 0 | 0 | 44 |
| 0 | 0 | 42 |
| 0 | 0 | 40 |
- [10pts] Fit a second order model including AB and A2 and evaluate the lack of fit.
- [10pts] Sketch a contour plot and identify the recommended operating conditions to minimize filtration time. Estimate the predicted filtration time under those conditions.
- [10pts] What operating conditions would you recommend if the objective is to operate the process at a mean filtration time very close to 46.
Problem 4 [10 points – EMIS 7377 students only] A manufacturer of cutting tools has developed two empirical equations for tool life in hours (y1) and for tool cost in dollars (y2). Both models are linear functions of steel hardness (x1) and manufacturing time (x2). The two equations are:
and both equations are valid over the range . Unit tool cost must be below $27.50 and expected tool life must exceed 12 hours for the product to be competitive.
- [2.5pts] Sketch contour plots for the two models.
- [2.5pts] Sketch the overlay plot
- [2.5pts] Is there a feasible set of operating conditions for this process?
- [2.5pts] Where would you recommend that the process be run?