Question

Design specifications require that a key dimension on a product measure 100 ± 10 units. A process being considered for producing this product has a standard deviation of four units.

a.	What can you say (quantitatively) regarding the process capability? Assume that the process is centered with respect to specifications. (Round your answer to 4 decimal places.)

Process capability

b.	Suppose the process average shifts to 92. Calculate the new process capability. (Round your answer to 4 decimal places.)

New process capability

c.	What is the probability of defective output after the process shift? (Use Excel's NORMSDIST() function to find the correct probability for your computed Z-value.Round "z" value to 2 decimal places and final answer to 4 decimal places.)

Probability of defective output

Answer 1

USL = 100 + 10 = 110 , LSL = 100 -10 = 90

a) CPk = Min ( USL - Mean) / 3 * Std deviation , ( Mean - LSL) / 3* standard deviation

= (10/(3*4) = 0.8333

b) New CPk = Min ( USL - Mean) / 3 * Std deviation , ( Mean - LSL) / 3* standard deviation

Mean = 92

= Min [( 110 - 92) / 12 , (92-90) / 12) ]

Min (18/12, 2/12) = Min ( 1.5 , 0.1666) = 0.1666

c) Z Value = (X - Mean) / Std. deviation

Now Probability of X = 90

Z-score (left tail) = (90-92) / 4 = -0.50 , Z-score (Right tail) = (110-92) / 4 = 4.5

P-value using NORMDIST(X, Mean, Std deviatiion, False) , NORDIST (90,92,4,False) = 0.6915

Probability of defective item = 1 - 0.6915 = 0.3085

OTHER METHOD

Z-score (left tail) = (90-92) / 4 = -0.50 , Z-score (Right tail) = (110-92) / 4 = 4.50

probability value at Z = -0.5 = 0.1915

probability value at Z = 4.5 = 0.5

Probability of value within 90 - 110 = 0.1915 + 0.5 = 0.6915

Probability of defective output = 1- 0.6915 = 0.3015