The number of runs required by a full-factorial design with three factors each at two levels is equal to

A full-factorial design examines every possible combination of the levels across all factors. The total number of experimental runs is calculated by the formula L^k, where 'L' is the number of levels for each factor and 'k' is the number of factors. In this scenario, there are three factors (k=3), and each factor is at two levels (L=2). Therefore, the total number of runs required is 2^3 = 2 × 2 × 2 = 8. This design allows for the estimation of all main effects and all interaction effects among the three factors.

A. 6 is incorrect. This result comes from incorrectly multiplying the number of factors by the number of levels (3 × 2), not exponentiating.
C. 9 is incorrect. This would be the number of runs for a design with two factors at three levels each (3^2), which is not the design described.
D. 12 is incorrect. This number does not correspond to a standard full-factorial calculation for the given parameters.

1. Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). John Wiley & Sons. In Chapter 6, "The 2^k Factorial Design," Section 6-2, the text states, "This is a 2^k factorial design... A complete replicate of this design requires 2 x 2 x ... x 2 = 2^k observations." For k=3, this is 2^3 = 8 runs (p. 205).

2. The ASQ Certified Quality Engineer Handbook (4th ed.). (2017). In Chapter 31, "Design and Analysis of Experiments," the section on Factorial Experiments explains that for a 2^k design, the number of treatment combinations is 2 multiplied by itself k times. For a 2^3 design, this is 8 combinations (p. 438).

3. Pennsylvania State University. (n.d.). STAT 503: Design of Experiments. Eberly College of Science. In Lesson 4: The 2^k Factorial Design, it is explicitly stated that "For k factors, each at 2 levels, the design is a 2^k factorial design... For k=3, we have a 2^3 design with 8 treatment combinations." Retrieved from https://online.stat.psu.edu/stat503/lesson/4.