Question

Control charts for x bar and s are maintained on a quality characteristic. The sample size is n=4. After 30 samples, we obtain

3030

∑ ¯x_i=12,870and∑ s_j=410

i=1i=1

A. Find the centerline and three sigma control limits for the s chart.

B. Assuming that both charts exhibit control,estimate the aprameters μ and σ.

Answer 1

Let S be a biased estimator of σ Therefore, E(S) C,0 n-1n-1 Here c,is a constant that is near, but not equal to 1 The centre line and three sigma control limits for S are expressed as follows CL C4 Compute the value of C, by using the above fomula 4-1/4-1 3 (0.5) = 0.82 Therefore, the value of C, for the given data is 0.82

The upper control limit for S chart C4 -13.667 +313.66 13.667+3(16667) (0.5724) 42.29 1-0.82 The central Iine for S chart: 13.667 The lower control limit for s chart C4 -13.667-3 15.667-0.82 13.667-3(16667) (0.5724) -14.95 Since the lower control limit is a negative number, we set LCL to be zero

Let us assume that the both charts exhibit control. It is given that the size ofthe sample n=4, X,-12870 and Σ,-410 Therefore, the estimate of the parameter μ is given by 2 - [12870 Therefore the value of parameter μ is 429 3 Estimator of σ from S chart is C4 2 1(410 30 = 13.667 Therefore, the estimator of σ is calculated as follows: 5 13.667 0.82 - 16.67