# From a process known to be in control, 6 samples of 4 units each were taken at random intervals and the units in the samples were weighed. The mean (Xbar) and range (R) for each of the six samples are given in the following table. Sample Mean Range 1 5.2 0.6 2 5.6 1.0 3 5.1 1.1 4 5.7 1.0 5 5.6 0.7 6 5.4 0.8 a. Calculate the 3-sigma Xbar-chart and R-chart control limits. b. Calculate the mean (Xbar) and range (R) for the following sample, which was taken from the same process at a later time. Item number 1 2 3 4 Weight 5.7 6.0 6.2 5.5 Based on this sample and the control chart limits that you calculated in part (a), is the process in control? Why or why not?

a) Grand mean bar{bar{X}} = ( 5.2 + 5.6 + 5.1 + 5.7 + 5.6 + 5.4 ) div 6

Grand mean bar{bar{X}} = 5.53

bar{R} = ( 0.6 + 1.0 + 1.1 + 1.0 + 0.7 + 0.8 ) div 6

bar{R} = 0.866

3-sigma Xbar-chart control limits

UCL= bar{bar{X}} + A_{2}bar{R}

The value of A2 from the table for a sample size of four equals 0.729

A2 = 0.729

UCL = 5.53 + 0.729 * 0.866

UCL = 6.16

LCL = 5.53 – 0.729 * 0.866

LCL = 4.90

3-sigma R – chart control limits

UCL= bar{R}D_{4}

UCL = 0.866 * 2.282

UCL = 1.98

LCL= 0

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b) Mean weight = ( 5.7 + 6 + 6.2 + 5.5 ) div 4

Mean weight = 5.85

The process is under control because the mean weight is less than the UCL i.e. 6.16

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