You are asked to analyze a drilling operation in which the o

You are asked to analyze a drilling operation in which the operator uses a 40mm drill bit to make a hole. Samples are taken at regular intervals in groups of 6. The x-bar and S values are calculated for each hole. The data appears reasonably normal. After 50 subgroups have been analyzed, we have: Compute the control limits for the x-bar and S control charts

Solution

S-bar = 75/50 = 1.5

X-(double bar) = 2000/50 = 400

n=6 ,

Factor C4 = sqrt(2/5) * (6/2 - 1 )! / (5/2 -1 )!

               = (sqrt(2/5) * 2 )/ ((1.5)(0.5)(1.77246))    -----------> [ (n/2)!=(n/2)(n/2?1)(n/2?2)?(1/2)sqrt(pi) ]

               = 0.95

The parameters for S chart :

UCL = s-bar + 3s-bar/C4 *sqrt ( 1-c4^4) = 1.5 + ((3*1.5) *sqrt ( 1-0.95^4) /0.95) = 3.54

Center Line = 1.5

LCL = s-bar - 3s-bar/C4 *sqrt ( 1-c4^4) = 1.5 - ((3*1.5) *sqrt ( 1-0.95^4) /0.95) = -0.54

The parameters for X-bar chart :

UCL = x(double bar) + (3s-bar) / (C4 * sqrt(n)) = 400 + ((3*1.5) / (0.95 * sqrt(6))) = 401.93

Center Line = 400

LCL = x(double bar) - (3s-bar) / (C4 * sqrt(n)) = 400 - ((3*1.5) / (0.95 * sqrt(6))) = 398.06