Design specifications that require a key dimension on a product measure 100 ± 10 units. A centered process is being considered for producing this product which has a standard deviation of four units.
(a) What can you say (quantitatively) regarding the process capability?
(b) Suppose the process average shifts to 92. Calculate the new process capability.
(c) What can you say about the process after the shift? Approximately what percentage of the items produced will be defective?

The Process capability of a process is determined using the capability index [tex]C_{pk}[/tex] . This capability index helps in determining whether the output of a process lies within the specification limits.

a. Given that

Mean, X = 100

Lower Specification Limit, LSL = 100 - 10 = 90

Upper Specification Limit, USL = 100 + 10 = 110

Standard Deviation, [tex]\alpha[/tex] = 4

Process Capability can therefore be calculated using the formula :

[tex]C_{pk}= min[\frac{X-LSL}{3\alpha }, \frac{USL-X}{3\alpha },]\\= min[\frac{(100-90)}{3*4 }, \frac{(110-100)}{3*4 },]\\=min[\frac{10}{12}, \frac{10}{12}]\\=min[0.8333, 0.8333] = 0.8333[/tex]

Hence, the process capability value for the process is 0.833

b. When the process average shifts to 92, the new process mean would be 92

X = 92

LSL = 90

USL = 110

Standard Deviation = 4

[tex]C_{pk} = min[\frac{X - LSL}{3\alpha }, \frac{USL - X}{3\alpha }]\\=min[\frac{(92-90)}{3*4}, \frac{(110-92)}{3*4}]\\=min(0.1667, 1.5)= 0.1667[/tex]

c. Probability of the defective unit after the shift :

We calculate the Z score of LSL like so :

[tex]Z_{LSL}=\frac{LSL - X}{\alpha }\\= \frac{90-92}{4}\\=- \frac {2}{4} = -0.5[/tex]

Using the "NORMDIST(Z)" function in Excel, we find the probability associated with [tex]Z_{LSL}[/tex] as 0.30854

Therefore the probability that the output will be less than 90 units is 0.30854