Samples of eight bolts are taken periodically, and their diameters (in mm) are m

Question

Samples of eight bolts are taken periodically, and their diameters (in mm) are measured. The following table presents the means, ranges, and standard deviations for 10 consecutive samples. (a) Using the data tabulated above, calculate the 3 sigma control limits for the R chart. Is the variance under control? (b) using the western Electric rules, is the process mean in control? If not, after which sample is it first detected to be out control? (c) Based on the sample range R, calculate the 3 sigma control limits for the X chart. Based on the 3 sigma limits, is the process mean in control?

Explanation / Answer

Note: for each variable we have to calculate a strd. desviation, then we have to multiplicate by 3 for 3S limits as shows below...

Sample 1 2 3 4 5 6 7 8 9 10 Mean X 9.99 10.02 10.1 9.9 9.92 10.05 9.97 9.93 10.01 9.87 9.976 R 0.28 0.43 0.16 0.26 0.22 0.4 0.08 0.48 0.25 0.29 0.285 s. 0.09 0.13 0.05 0.09 0.07 0.15 0.03 0.15 0.09 0.1 0.095 X new mean R new mean s new mean 1 9.99 9.976 0.000196 0.004644 0.28 0.285 0.000025 0.013605 0.09 0.095 0.000025 0.001425 2 10.02 0.001936 0.43 0.021025 0.13 0.001225 3 10.1 0.015376 0.16 0.015625 0.05 0.002025 4 9.9 0.005776 0.26 0.000625 0.09 0.000025 5 9.92 0.003136 0.22 0.004225 0.07 0.000625 6 10.05 0.005476 0.4 0.013225 0.15 0.003025 7 9.97 3.6E-05 0.08 0.042025 0.03 0.004225 8 9.93 0.002116 0.48 0.038025 0.15 0.003025 9 10.01 0.001156 0.25 0.001225 0.09 0.000025 10 9.87 0.011236 0.29 2.5E-05 0.1 0.000025 0.046440 0.13605 0.01425 3SX Sigma = SQRT(.004644) 0.0681469 3Sigma= 0.2044407 3SR Sigma = SQRT(.013605) 0.116640473 3Sigma= 0.34992142 3SS Sigma = SQRT(.001425) 0.037749172 3Sigma= 0.11324752 a) 3SR= .3499 The variance between the data and 3S limits shows that the process is under control because only two measures are on top of 3S limits b) Western Electric rules, stablish that less variation on measures is a better process, then for R chart, there is not a big variation between data, only two presents a higher level upon 3S limits c) 3SX = .2044 taking R as a refference point compared with 3SX limits, we can observe that the most measures are upon the 3S limits, then the process mean is not under control, there is eight measures upon 3S limits .

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