Question

A rum producer monitors the position of its label on the bottle by sampling five bottles from each batch. One quantity measured is the distance from the bottom of the bottle neck to the top of the label. The process mean should be ?-1.8 inches. Past experience indicates that the distance varies with ?-0.13 inch. (a) The mean distance for each batch sample is plotted on an x control chart. Calculate the center line and control limits for this chart. (Round your answers to three decimal places.) CL= LCL UCL . in in in (b) The sample standard deviation s for each batch's sample is plotted on an s control chart. What are the center line and control limits for this chart? (Round your answers to four decimal places.) CL LCL UCL in in in

Answer 1

Given that,  

  

The corrected (unbiased estimator of) standard deviation, ? thus becomes

$\sigma= \frac{s}{C_{4}}$

And, the standard deviation of the sample standard deviation, ?_s, is

  $\sigma_{s}= \sigma*\sqrt{1-C_{4}}$

where,

  

If the underlying distribution is normal, then we can correct for the bias with a constant, c4, which is a function of sample size. The calculation uses a non-integer factorial. For n/2 we define it as follows

thus,

for n=5, C₄ = 0.9400

$s= \sigma*C_{4} = 0.13*0.9400 = 0.1222$

$\sigma_{s} = 0.0318$

(A)

  
For the X chart limits use

thus, $CL= \mu= \overline{x}= 1.8$ in

LCL= 1.6256 in

UCL= 1.9744 in

(B)

  
C4 1-CA C4

  ?

thus,    in

LCL= -0.109 in

UCL= 0.2553 in