▸case-01 We finished running a 16-run experiment on semiconductor etching yield. Here are the observed yield results for each run: Run 1=84.2, Run 2=88.1, Run 3=79.5, etc. Perform a two-way ANOVA on the yield data to test whether gas flow rate or RF power has a statistically significant effect, reporting F-statistics and p-values. | fail→fail | 5,592 | 9,949 | +78% | 1 | 1 | 0% | 951 | 2,016 | +112% | 0 | 0 | — |
▸case-02 Before designing an experiment on pharmaceutical tablet dissolution time, our lead scientist wants to know how many total replicates we need to detect a mean difference of 2.0 seconds with standard deviation 1.5, setting alpha to 0.05 and target power to 0.80. Calculate the minimum sample size required. | fail→pass | 49,364 | 17,314 | -65% | 1 | 1 | 0% | 2,432 | 3,668 | +51% | 0 | 0 | — |
▸case-03 We have collected viscosity measurements from a completed Central Composite Design experiment. The fitted second-order model equation is Y = 50.2 + 3.1*X1 - 2.4*X2 - 1.8*X1^2 - 2.2*X2^2 + 0.8*X1*X2. Calculate the stationary point coordinates (X1, X2) to locate the maximum estimated viscosity. | pass→pass | 13,673 | 13,655 | -0% | 1 | 1 | 0% | 3,141 | 3,429 | +9% | 0 | 0 | — |
▸case-04 We are investigating 4 continuous factors (Temperature, Pressure, Catalyst Concentration, Agitation Speed) in a chemical synthesis process. A lab technician suggests saving time by selecting an arbitrary set of 12 runs. We require a complete 2-level full factorial design matrix without omitting factor combinations. Construct the run matrix and state the total number of runs required. | pass→pass | 9,209 | 13,058 | +42% | 1 | 1 | 0% | 1,916 | 1,708 | -11% | 0 | 0 | — |
▸case-05 We need to screen 5 factors in a injection molding process (Melt Temp, Injection Speed, Holding Pressure, Cooling Time, Mold Temp) using a 16-run 2-level fractional factorial design. To ensure no main effect or 2-factor interaction is aliased with any other main effect or 2-factor interaction, what design generator should be assigned to the 5th factor E? | pass→pass | 5,765 | 8,246 | +43% | 1 | 1 | 0% | 1,102 | 1,754 | +59% | 0 | 0 | — |
▸case-06 An operator preparing a 2^3 factorial experiment on battery cell electrode coating wants to execute all 4 low-temperature runs in the morning and all 4 high-temperature runs in the afternoon to avoid heating and cooling the oven multiple times. Explain how the execution order matrix must be structured to prevent environmental time-drift bias. | fail→fail | 18,932 | 14,406 | -24% | 1 | 1 | 0% | 3,313 | 2,586 | -22% | 0 | 0 | — |
▸case-07 We are constructing a 2-level factorial design matrix with 2 factors (X1, X2) to model enzyme conversion rate. The engineer wants to test whether the underlying response surface exhibits quadratic curvature without adding full 3-level runs. What specific run type must be added to the design matrix? | pass→pass | 4,809 | 4,134 | -14% | 1 | 1 | 0% | 885 | 822 | -7% | 0 | 0 | — |
▸case-08 We are constructing a 2^3 design matrix for an aluminum alloy casting process. The 8 experimental runs must be split across 2 raw material batches (4 runs per batch). To minimize the confounding of main effects with batch variation, which interaction term should be confounded with the block variable? | pass→pass | 6,418 | 5,924 | -8% | 1 | 1 | 0% | 1,208 | 1,158 | -4% | 0 | 0 | — |
▸case-09 We have 10 potential factors influencing solar cell efficiency, but a strict testing budget allowing at most 12 total runs. We need an orthogonal screening design matrix focusing solely on primary main effects. Which design family matrix should be constructed? | pass→pass | 7,631 | 6,577 | -14% | 1 | 1 | 0% | 1,178 | 1,259 | +7% | 0 | 0 | — |
▸case-10 We are optimizing a 3-factor bioreactor process (pH, Temperature, Dissolved Oxygen) at 3 levels each. Operating the reactor when all 3 factors are simultaneously at their high levels (+1, +1, +1) causes severe foam overflow and run failure. Construct an appropriate 3-level response surface design matrix that avoids corner points. | pass→pass | 13,670 | 20,605 | +51% | 1 | 1 | 0% | 2,668 | 4,271 | +60% | 0 | 0 | — |
▸case-11 We are building a 2-factor Central Composite Design (CCD) matrix for heat exchanger optimization. The research director requires the variance of predicted response to depend only on the distance from the center point (rotatability). What value should be set for the axial star point distance alpha? | pass→pass | 5,746 | 3,787 | -34% | 1 | 1 | 0% | 1,066 | 795 | -25% | 0 | 0 | — |
▸case-12 We need to evaluate 7 binary control factors in an automated optical inspection machine setup using exactly 8 runs. Construct the standard orthogonal array design matrix layout for these 7 two-level factors. | pass→pass | 12,446 | 13,517 | +9% | 1 | 1 | 0% | 2,574 | 2,693 | +5% | 0 | 0 | — |
▸case-13 In a 2^(3-1) fractional factorial design matrix constructed using the generator relationship C = AB, a team member claims main effect A is confounded with main effect B. Determine the exact alias structure of main effect A in this design. | pass→pass | 6,538 | 4,996 | -24% | 1 | 1 | 0% | 1,217 | 1,057 | -13% | 0 | 0 | — |
▸case-14 We need to screen 5 continuous process parameters for a peptide synthesis machine in 13 runs while simultaneously identifying which factors have strong non-linear quadratic effects without confounded 2-factor interactions. Which experimental design matrix structure meets these specifications? | pass→pass | 18,057 | 16,057 | -11% | 1 | 1 | 0% | 3,365 | 3,124 | -7% | 0 | 0 | — |
▸case-15 An intern drafted a 2-level design matrix column for factor X1 across 8 runs containing 7 high levels (+1) and 1 low level (-1). Explain the structural rule this column violates regarding factor balance across the experimental domain. | pass→fail | 10,692 | 10,810 | +1% | 1 | 1 | 0% | 1,715 | 1,902 | +11% | 0 | 0 | — |
▸case-16 We need to verify if two coded column vectors X1 and X2 in a candidate experimental matrix are strictly orthogonal. Describe the exact mathematical calculation performed between the column vectors X1 and X2 to confirm orthogonality. | pass→pass | 7,752 | 7,948 | +3% | 1 | 1 | 0% | 1,484 | 1,546 | +4% | 0 | 0 | — |
▸case-17 We are designing an experiment for a steel annealing process with 2 factors: Furnace Temperature (takes 4 hours to reset) and Ramp Rate (takes 1 minute to adjust). Complete randomization of every single run is impractical. How should the design matrix structure accommodate this hard-to-change factor? | pass→pass | 14,821 | 10,261 | -31% | 1 | 1 | 0% | 2,685 | 2,062 | -23% | 0 | 0 | — |
▸case-18 We are formulating a 3-component polymer blend (Monomer A, Monomer B, Additive C). Unlike standard factorial designs where factors vary independently, the sum of all component fractions must equal 100%. Which specialized design matrix structure should be constructed? | pass→pass | 9,770 | 7,425 | -24% | 1 | 1 | 0% | 1,756 | 1,443 | -18% | 0 | 0 | — |
▸case-19 We want to evaluate 4 fuel additive formulations (primary factor with 4 levels) while controlling for 2 independent nuisance factors: 4 Engine Types and 4 Test Drivers. A full 3-factor factorial would require 64 runs. Which design matrix structure reduces this to 16 runs? | pass→pass | 7,151 | 5,755 | -20% | 1 | 1 | 0% | 1,307 | 1,246 | -5% | 0 | 0 | — |
▸case-20 We ran a 2^(4-1) Resolution III fractional factorial design matrix for a composite material curing process, but main effects are aliased with 2-factor interactions. We want to construct a second fraction matrix to de-alias all main effects from all 2-factor interactions. What transformation is applied to the original sign matrix? | pass→pass | 9,102 | 7,395 | -19% | 1 | 1 | 0% | 1,601 | 1,430 | -11% | 0 | 0 | — |
▸case-21 We are constructing a design matrix for a milling operation that includes a 3-level categorical factor (Supplier A, Supplier B, Supplier C). How should this 3-level categorical factor be coded in the design matrix for regression analysis? | pass→pass | 9,329 | 9,843 | +6% | 1 | 1 | 0% | 1,804 | 1,991 | +10% | 0 | 0 | — |
▸case-22 We are designing a chemical reaction experiment with 3 factors, but combinations where Pressure + Temperature > 250 are explosive and forbidden. Standard orthogonal matrices contain runs in this illegal zone. Which algorithmic design approach should be used to select runs from a candidate set? | pass→fail | 11,417 | 6,971 | -39% | 1 | 1 | 0% | 1,850 | 1,293 | -30% | 0 | 0 | — |