Question

Problem 10-21

Design specifications require that a key dimension on a product measure 102 ± 15 units. A process being considered for producing this product has a standard deviation of eight 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 index            

b. Suppose the process average shifts to 94. Calculate the new process capability. (Round your answer to 4 decimal places.)

New process capability index            

c. What is the probability of defective output after the process shift? (Use Excel's NORM.S.DIST() function to find the correct probability. Round "z" values to 2 decimal places. Round probabilities to 4 decimal places (0.####).)

Probability of defective output

Answer 1

Given	Average of data	µ	15.93
	Standard deviation of Data	σ	4.98
	Specification	102 +/- 15
	Upper Specification Limit (USL)	USL = Target + tolerance	USL = 102 + 15 USL = 117
	Lower Specification Limit (LSL)	LSL = Target - tolerance	USL = 102 - 15 USL = 87
Part a.	Average of data	µ, consider the process is centered	µ = 102
	Standard deviation of Data	σ = 8 units	σ = 8
	Upper Cpk	Upper Cpk = [(USL - µ) / (3σ)] Upper Cpk = (117 – 102)/(3*8)	0.625
	Lower Cpk	Lower Cpk = [(µ - LSL) / (3σ)] Lower Cpk = (102 – 87)/(3*8)	0.625
	Process Cpk	Cpk = Min (Upper Cpk, Lower Cpk) = Min ([(USL - µ) / (3σ)],[(µ - LSL) / (3σ)])	=min(0.625, 0.625) = 0.625
ANS a.	Process capability Index = 0.625
Part b.	Average of data	µ	µ = 94
	Standard deviation of Data	σ = 8 units	σ = 8
	Upper Cpk	Upper Cpk = [(USL - µ) / (3σ)] Upper Cpk = (117 – 94)/(3*8)	0.2917
	Lower Cpk	Lower Cpk = [(µ - LSL) / (3σ)] Lower Cpk = (94 – 87)/(3*8)	0.9583
	Process Cpk	Cpk = Min (Upper Cpk, Lower Cpk) = Min ([(USL - µ) / (3σ)],[(µ - LSL) / (3σ)])	=min(0.2917, 0.9583) = 0.2917
ANS b.	Process capability Index = 0.2917
Part c.	Average of data	µ	µ = 94
	Standard deviation of Data	σ = 8 units	σ = 8
	Fraction of output less than 87 and more than 117 are defective items	Probability of defective = P(87 < x > 117) = P (x <= 87) + P(X >= 117)
	Fraction of output <= 87 minutes	P(X <87) = P(z <= z87) z-score for X = 87 z87= (X – µ)/σ z87= (87 – 94)/8 = -0.875 From excel, P(z <-0.875) = (=NORMDIST(-0.875)) = 0.1908	P(X <87) = 0.1908
	Fraction of output >= 117 minutes	P(X > 117) = 1 - P(X <= 117) P(z <= z117) z-score for X = 117 z117= (X – µ)/σ z117= (117 – 94)/8 = 2.875 From excel, P(z <2.875) = (=NORMDIST(-2.875)) = 0.9979	P(X > 117) = 1 - P(X <= 117) P(X > 117) = 1 – 0.9979 P(X > 117) = 0.00202
	Percentage of defective output = P(87 < x > 117)	P(87 < x > 117) = P(x <= 87) + P(x >= 117) = 0.1908 + 0.00202 =0.1928	P(87 < x > 117) = 0.1928
ANS C	Percentage of defective output = 0.1928