Final answer:
To calculate CpK, we use the process capability, along with the upper and lower specification limits. After finding the standard deviation, we use it and the mean to compute CpK. In this case, assuming the mean is centered, the CpK value matches the given Cp of 1.58.
Step-by-step explanation:
To calculate the capability index (CpK), we need the process capability (Cp), the upper specification limit (USL), the lower specification limit (LSL), and the process mean (μ). The formula for CpK is min[(USL - μ) / (3σ), (μ - LSL) / (3σ)]. Given that the Director of Marketing for the business has established the pricing guidelines to be between $1.2 (LSL) and $2.4 (USL), the local sales representative should operate within these limits. With a Cp of 1.58, we can find the standard deviation (σ) using the formula Cp = (USL - LSL) / (6σ), which simplifies to σ = (USL - LSL) / (6Cp). Using the given limits for pricing guidelines, let's calculate the standard deviation and subsequently the CpK value.
Standard deviation (σ) = (USL - LSL) / (6 Cp) = (2.4 - 1.2) / (6 * 1.58) = 1.2 / 9.48 = 0.126582278.
Assuming the process mean (μ) is at the midpoint of the LSL and USL, μ = (LSL + USL) / 2 = (1.2 + 2.4) / 2 = 1.8.
Now we can calculate CpK:
CpK = min[(USL - μ) / (3σ), (μ - LSL) / (3σ)]
= min[(2.4 - 1.8) / (3 * 0.126582278), (1.8 - 1.2) / (3 * 0.126582278)]
= min[0.6 / 0.379746835, 0.6 / 0.379746835]
= min[1.581027667, 1.581027667].
As both minimum calculations give the same result, CpK = 1.58.