Chapter 2: Q13-5OQ (page 339)

Question:Design specifications require that a key dimension on a product measure 100 6 10 units. A process being considered for producing this product has a standard deviation of four units.
What can you say (quantitatively) regarding the process capability?
Suppose the process average shifts to 92. Calculate the new process capability.
What can you say about the process after the shift? Approximately what percentage of the items produced will be defective?

Short Answer

Expert verified

Answer

$Usethisformula: C_{pk} = \min (\frac{UTL - X - LTL}{3 σ (3 σ)})$

Standard deviation is important because it helps us to understand how spread the values are in the given dataset. In real life, standard deviations are used by the school or universities professors to calculate the spread of the student’s test scores among them.

Step by step solution

Step-by-Step Solution Step 1: (a) Calculate the process capability

UTL= 110,

X= 100,

LTL= 90,

Sigma = 4

$\begin{array}{rcl} \begin{matrix} C_{pk} = \min (\frac{110 - 100, 100 - 90}{3 (4), 3 (4)}) \\ = \min (0.8333, 0.8333) \\ C_{pk} = 0.8333 \end{matrix} \end{array}$

(b) Calculate the new process capability

UTL= 110,

X= 92,

LTL = 90,

Sigma = 4

$\begin{matrix} C_{pk} = \min (\frac{110 - 92, 92 - 90}{3 (4), 3 (4)}) \\ = \min (1.5000, 1.667) \\ C_{pk} = 0.1667 \end{matrix}$

(c) Approx. percentage of defected items

Many defects were produced. Assume a normal distribution, that the left tail, {Prob (x<90)} is $z = \frac{(90 - 92)}{4} = - 0.50$ , which corresponds to a probability of 0.3085.

The right tail, {Prob(x>110) = 1-Prob (x<110)} is $z = \frac{(110 - 92)}{4} = 4.5$ , which corresponds to a probability of 0.1-0.9999966 = 0.0000034.

The above calculation assumes that 0.3085+0.0000034 = 0.3085 or approximately 31% are outside the specification limits.

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Short Answer

Step by step solution

Step-by-Step Solution Step 1: (a) Calculate the process capability

(b) Calculate the new process capability

(c) Approx. percentage of defected items

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