Question

An in-control process has an estimated standard deviation of 2 mm. The specification limits of the component being processed are 120 ± 8 mm. When the process mean shifts to 118 mm, the values of the process capability indices, \( C_p \) and \( C_{pk} \), respectively, are

• A. 1.000, 1.667
• B. 1.333, 1.667
• C. 1.333, 1.000
• D. 1.000, 1.000

Hint:

For process capability indices, \( C_p \) measures the spread of the process relative to the specification limits, while \( C_{pk} \) also accounts for the process mean shift.

Answer 1

The Correct Option is C

The process capability indices \( C_p \) and \( C_{pk} \) are defined as follows: \[ C_p = \frac{USL - LSL}{6 \sigma} \] \[ C_{pk} = \min \left( \frac{USL - \mu}{3 \sigma}, \frac{\mu - LSL}{3 \sigma} \right) \] where:
- \( USL = 120 + 8 = 128 \) mm is the upper specification limit,
- \( LSL = 120 - 8 = 112 \) mm is the lower specification limit,
- \( \sigma = 2 \) mm is the standard deviation of the process,
- \( \mu = 118 \) mm is the shifted process mean.
Step 1: Calculate \( C_p \) \[ C_p = \frac{128 - 112}{6 \times 2} = \frac{16}{12} = 1.333. \] Step 2: Calculate \( C_{pk} \) For the shifted mean \( \mu = 118 \), the closest specification limit is the lower specification limit: \[ C_{pk} = \frac{118 - 112}{3 \times 2} = \frac{6}{6} = 1.000. \] Thus, the correct answer is (C).