Question

Calculate the OC curve of the 2.5-sigma C-chart to control the number of defects per engine assembly at nine. Use c values of 3, 5, 7, 9, 10, 15, and 20.

Answer 1

You need a cumulative poisson table for this.

So we have a c chart with 2.5 as the sigma value so we need to find the control limits first.

The control limits of a 2.5 sigma chart would be found by

c was chosen as 9 because the number of defects per engine assembly has to be kept at 9 and it was taken as the mean value of the number of defects for n number of engines checked for inspection.

Now, to calculate c chart with varying values of c, we will use c values in the x axis and probability of acceptance, as the y axis.

The formula required to calculate probability of aceptance would be:-

where c is the acceptance number and c1 is any acceptance number other than the one for which we are designing which is 9 in our case.

This tells us that probability of acceptance is the difference between the probability value for the case when the number of defects is less than or equal to 16 (greatest number of defects less than 16.5 which is the upper limit) and the probability for the case when the number of defects is less than or equal to 1 (1 is the greatest number of defects possible that is less than 1.5).

Thus an engine is accepted according to the probability of acceptance when the number of defects fall within the control limits

Now these probability values are calculated from the cumulative Poisson table whose columns give the value for a particular mean value (). For c chart, . The rows of cumulative poisson table give values for different values of c.

For c = 3, (the value of c1 becomes 3 in the formula for Pa)

Finding out the value from cumulative poisson table, for first term in the column header () look for the value 3 and in the row header ( or ), look for the value 16. and the cell that contains the corresponding value contains your answer. For the second term you again look for c in column but 1 in row as C <= 1, you take the value in the corresponding cell of the table. Then subtract the two and you get your probability of acceptance for c = 3.

Similarly for c = 5, we replace c1 with 5 and use the table to find out values

Similarly calculate Pa for every value of c as required and these values are summarized in the table below:-

	3	5	7	9	10	15	20
	0.801	0.96	0.992	0.987	0.974	0.664	0.221

Using these points you can plot vs. graph which wil be your OC curve.

Certain points to notice here, you can see that as the acceptance number increases to 15 or 20 which would contain that many defects, the probability of acceptance is rightfully reduced sharply thus indicating a good sampling plan.