Final answer:
The calculated process capability index (Cp) is 1.4167, indicating that the process has the potential to meet its specification limits. The process capability ratio (Cpk) is also 1.4167, assuming the process mean is perfectly centered between the specification limits, which demonstrates that the process is capable as long as the mean remains centered.
Step-by-step explanation:
The process capability indices, including Cpk and Cp, are measures of a process's ability to produce output within specified limits. In this case, we are considering bags of Jelly Belly jellybeans with a defined upper specification limit (USL) of 363 jellybeans, a lower specification limit (LSL) of 329 jellybeans, and a process standard deviation (σ) of 4 jellybeans.
Calculating the Cp index
The Cp index (Process Capability Index) measures a process's potential capability and is calculated using the formula:
Cp = (USL - LSL) / (6σ)
For our case:
Cp = (363 - 329) / (6 * 4)
Cp = 34 / 24
Cp = 1.4167
Calculating the Cpk index
The Cpk index (Process Capability Ratio) is the minimum of two values that measure how centered the process mean is within the specification limits. We calculate them as follows:
Cpk lower = (mean - LSL) / (3σ)
Cpk upper = (USL - mean) / (3σ)
Since the mean is not provided, we will assume that it is centrally located, i.e., (USL + LSL) / 2 = (363 + 329) / 2 = 346.
Cpk lower = (346 - 329) / (3 * 4)
Cpk lower = 17 / 12
Cpk lower = 1.4167
Cpk upper = (363 - 346) / (3 * 4)
Cpk upper = 17 / 12
Cpk upper = 1.4167
Therefore, assuming the process mean is centered, Cpk = 1.4167.
These indices suggest that the process is capable as long as the mean is centered. A Cp and Cpk greater than 1.0 typically indicates that the process is capable of meeting specification limits.