Question

The refrigeration process at the Brownsville plant of a national food processing firm is known to be in control and normally distributed with a mean of 38.21 degrees Fahrenheit and a standard deviation of 0.15 degrees Fahrenheit. In accordance with the company's policy, the refrigeration process specification is for temperature to be 38.25 ± 0.5 degrees Fahrenheit. The minimum required capability for the process is 1.66.

Calculate Cp and Cpk. Is the process capable? Is the process centered?

Answer 1

The process is not capable as both Cp and Cpk are less than the minimum required capability of 1.66 and the process is not centered as Cp and Cpk are not equal.

Step-by-step explanation:

The capability of a process is determined by Cp and Cpk. Both represent the ability of a process to meet specification requirements. They're calculated using the ratio of the specification range to the process variation (measured using the standard deviation).

First, let's calculate Cp. The formula for Cp is (USL - LSL) / 6σ, where USL and LSL are the upper and lower spec limits, and σ is the standard deviation. In this case, we know that the USL is 38.25 + 0.5 = 38.75 and the LSL is 38.25 - 0.5 = 37.75. The standard deviation is 0.15. Therefore, Cp = (38.75 - 37.75) / (6 * 0.15) = 1.11, which is less than the minimum required capability of 1.66, indicating that the process is not capable.

Next, let's calculate Cpk. The formula for Cpk is the minimum of [ (USL - μ) / 3σ , (μ - LSL) / 3σ ], where μ is the mean. In this case, the mean is 38.21. Therefore, Cpk = min([ (38.75 - 38.21) / (3 * 0.15) , (38.21 - 37.75) / (3 * 0.15) ]) = min(1.2, 1.02) = 1.02. This is also below the desired capability of 1.66, indicating that the process is not capable. The fact that Cp and Cpk are not equal also suggest the process is not centered within the specification limits.