Question

An x ̅ chart with three-sigma limits has parameters as follows: UCL = 109 Center Line = 101 LCL = 100 n = 5 Find the ARL1 for the chart. (10 points) What information does ARL convey that the statistical error probabilities do not?

Answer 1

SOLUTION:

From given data,

The required information is given by

chart with three-sigma limits has parameters

Center Line = 101

UCL = 109

LCL = 100

n = 5

To find the ARL1 for the chart.

Here , 3 = 4

= 4/3 = 1.33

for  3 level with control limit

= 0.0027

Now,

ARL = 1/

= 1/0.0027

= 370

ARL1 =  370 points

That is the control is even , here we accept the 1 point to plot outside the control limits for every 370 points

When the process in control

ARL = 1/

Where

= Type - 1 error

When the process in out of control

ARL = 1/ (1- )

Here is the probability of type ii error and is the probability of  Type - 1 error