Averages and ranges of samples of size n = 5 are shown in the accompanying table

ID: 3178657 • Letter: A

Question

Averages and ranges of samples of size n = 5 are shown in the accompanying table. The data pertain to the depth-to-shoulder measurement of glove boxes manufactured by an American car company. The measurements are in inches. From the first 25 samples, set up and interpret X and R control charts. Indicate whether or not control is evident. If the chart shows the process to be in a state of control, plot the next 5 samples and comment on the results. Sample X^bar Range Sample X^bar Range 1 0.4402 0.015 16 0.4362 0.015 2 0.4340 0.018 17 0.4380 0.019 3 0.4448 0.018 18 0.4350 0.008 4 0.4432 0.006 19 0.4378 0.011 5 0.4428 0.008 20 0.4384 0.009 6 0.4382 0.010 21 0.4392 0.006 7 0.4358 0.011 22 0.4378 0.008 8 0.440 0.019 23 0.4362 0.016 9 0.4366 0.010 24 0.4348 0.019 10 0.4338 11 0.4360 0.011 26 0.4366 0.014 12 0.4402 0.007 27 0.4346 0.019 13 0.4332 0.008 28 0.4374 0.015 14 0.4356 0.017 29 0.4339 0.024 15 0.4314 0.010 30 0.4368 0.014

Explanation / Answer

Back-up Theory

Grand Average, xdouble bar = (sum of xi bar over i)/k

Average Range, R bar = (sum of Ri over i)/k

LCL (Lower Control Limit) for x bar chart: x double bar - (A2).(R bar)

UCL (Upper Control Limit) for x bar chart: x double bar + (A2).(R bar)

LCL (Lower Control Limit for R chart): (D3).(R bar)

UCL (Upper Control Limit) for R chart: (D4).(R bar)

Sub-group size, n = 5, Number of samples, K = 25

Values of A2, D3, D4 and d2 are available in Table of Control Chart Constants

Now, to work out solution,

Part (a)

Xdouble bar = 0.4382

Rbar = 0.0122

A2(5) = 0.577

D3(5) = 0

D4(5) = 2.114

Central Line for Xbar Chart: 0.4382

LCL for Xbar Chart: 0.4312

UCL for Xbar Chart: 0.4452

Central Line for R Chart: 0.0122

LCL for R Chart: 0

UCL for R Chart: 0.0258

ANSWER

Part (b)

For the next 5 samples, xbar are: 0.4366, 0.4344, 0.4374, 0.4339 and 0.4368

All the values are within the limits.

The respective ranges are: 0.014, 0.019, 0.015, 0.024 and 0.014

All these values are also within limits.

Additional observations: While the range values are distributed above and below the central line, all xbar values are below the central line. The latter phenomenon is indicative of a possible downward shift in the setting, which needs to be corrected.