Refer to Table S6.1- Factors for Computing Control Chart Limits (3 sigma) for th

Question

Refer to Table S6.1- Factors for Computing Control Chart Limits (3 sigma) for this problem. Sampling 4 pieces of precision-cut wire (to be used in computer assembly) every hour for the past 24 hours has produced the following results Hour X R Hour x R Hour x R Hour x 1 3.35" 0.76" 2 3.00 1.23 3 3.32 1.43 4 3.49 1.21 5 3.17 1.17 6 2.96 0.42 7 2.95" 0.58" 13 3.21 0.90" 194.51" 1.56" 8 2.55 1.13 14 2.83 1.36 20 2.89 1.09 9 2.92 0.66 15 3.12 1.11 21 2.65 1.08 10 2.95 1.33 16 2.74 0.50 22 3.38 0.46 11 2.83 117 17 2.96 1.4323 12 2.97 0.40 18 2.74 1.3424 3.04 2.64 1.63 1.02 Based on the sampling done, the control limits for 3-sigma x chart are (round all intermediate calculations to three decimal places before proceeding with further calculations): Upper Control Limit (UCLinches (round your response to three decimal places). Lower Control Limit (LCL?)-inches (round your response to three decimal places). Based on the x-chart, the wire cutting process has been The control limits for the 3-sigma R-chart are (round all intermediate calculations to three decimal places before proceeding with further calculations) Upper Control Limit (UCLR)inches (round your response to three decimal places).

Explanation / Answer

Solution:

From the given information, the mean (X-double bar) and range (R-bar) is calculated as below;

X-double bar = Sum of observations (x-bar) / Total number of observations

X-double bar = 73.17 / 24

X-double bar = 3.049

R-bar = Sum of observations (R) / Total number of observations

R-bar = 24.97 / 24

R-bar = 1.040

(a) The Upper Control Limit (UCLx) and Lower Control limit (LCLx) for the 3-sigma x-bar chart is calculated as,

UCLx = X-double bar + (A2 x R-bar)

LCLx = X-double bar - (A2 x R-bar)

From the table of factors for computing the control chart limits (3-sigma),

For n = 4 (4-pieces of precision cut wire)

A2 = 0.729

Putting these values in the above formulas, we get,

UCLx = 3.049 + (0.729 x 1.040)

UCLx = 3.807 inches

LCLx = 3.049 - (0.729 x 1.040)

?LCLx = 2.291 inches

Based on the x-bar chart, the wire cutting process has been out of control, because the observation at 19th hour (4.51) lie outside the computed upper control limit and lower control limit.

(b) The Upper Control Limit (UCLr) and Lower Control limit (LCLr) for the 3-sigma R-chart is calculated as,

UCLr = D4 x R-bar

LCLr = D3 x R-bar

From the table of factors for computing the control chart limits (3-sigma),

For n = 4 (4-pieces of precision cut wire)

D3 = 0, D4 = 2.282

Putting these values in the above formulas, we get,

UCLr = 2.282 x 1.040

UCLr = 2.373 inches

LCLr = D3 x R-bar

LCLr = 0 x 1.040

LCLr = 0 inches

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