Question

1. Design X-bar and R charts a control chart with "standards given" as an aimed-at mean of Xo = 4.0, Sigma = .0033, and Subgroup size = 5. It is not necessary to sketch the control-chart since we have no points to put on it. Just specify, CL, UCL and LCL. That is the design.

2. Then find the probability of in-control nonconformance given, USL = 4.00995 and LSL = 3.99005

3. Also, by theory, what is the Probability of an out-of-control point when the process is in-control (type-II error). No math required but feel free to do a proof.

4. Specify the long term, in-control ARLo.

5. Then find the long term process capability ratio, Cp.

Answer 1

Given data

x_bar =4.0

sigma=0.0033

  n=5 ; m=3

For X_bar Chart

LCL = x_bar - m*(sigma/sqrt(n)) = 4.0-3*(0.0033/sqrt(5))=4.0-3*0.001476 = 4.0 - 0.004427 = 3.995573

CL = X_bar = 4.0

UCL = x_bar + m*(sigma/sqrt(n)) = 4.0 + 3*(0.0033/sqrt(5))=4.0 + 3*0.001476 = 4.0 + 0.004427 = 4.004427

UCL and LCL are within specification limit, which means it is in control process.

CP = (UCL-LCL)/(6*SIGMA) = (4.004427-3.995573)/(6*0.0033) = 1.005

For probability,

ZLCL = (X₀ -LCL)/SIGMA

Z_LSL =  (X₀ -LSL)/SIGMA

P = { AREA UNDER X₀-BAR TO LSL - AREA UNDER X₀-BAR TO LCL}

SIMILARLY FOR USL AND LSL.