Question

Auto pistons at Wemming​ Chung's plant in Shanghai are produced in a forging​ process, and the diameter is a critical factor that must be controlled. From sample sizes of 10 pistons produced each​ day, the mean and the range of this diameter have been as​ follows:

Day	Mean x (xx)	Range R (mm)
1	158.9	4.4
2	151.2	4.6
3	151.6	3.9
4	155.5	5.0
5	156.6	4.5

a. What is the value of the mean of x?

b. what is the mean of R?

c. what are the UCL_x and LCL_x using 3 sigma?

d. what are the UCLR and the LCLR using 3 sigma?

e. if the true diameter mean should be 115 mm and you want this as your center (nominal) line, what are the new UCL_x and LCL_x ?

Answer 1

Day	Mean x (xx)	Range R (mm)
1	158.9	4.4
2	151.2	4.6
3	151.6	3.9
4	155.5	5
5	156.6	4.5
	154.760	4.480
	X-barbar	R bar

Answer a: Mean of x= 154.760

Answer b: Mean of y= 4.480

D4, D3 and A2 are taken from table of factors computing 3 sigma control limits

D4 (n=10)=	1.777
D3 (n=10)	0.223
control limits for Range,
CL or R-bar= Average of all sample Range	4.480
UCL=R-bar*D4	7.961
LCL=R-bar*D3	0.999
A2 (n=10)	0.308
control limits for X bar,
CL or Xbarbar	154.760
UCL=Xbarbar+ (A2)*Rbar	156.140
LCL=Xbarbar- (A2)*Rbar	153.380

Answer c: UCLx= 156.140, LCLx= 153.380

Answer d:  UCLr= 7.961, LCLr= 0.999

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