Question

3.32. Quality Control. An automatic machine in a small factory produces metal parts. Most of the time (90% according to long-term records), it produces 95% good parts, while the remaining parts have to be scrapped. Other times, the machine slips into a less productive mode and only produces 70% good parts. The foreman observes the quality of parts that are produced by the machine and wants to stop and adjust the machine when she believes that the machine is not working well. Suppose that the first dozen parts produced are given by the sequence

s u s s s s s s s u s u

where s is satisfactory and u is unsatisfactory. After observing this se- quence, what is the probability that the machine is in its productive state? If the foreman wishes to stop the machine when the probability of “good state” is under 0.7, when should she stop it?

Answer 1

Now let us check for which C the above condition satisfying.

n	12		X	P(X=x)	P(X≤x)
p	0.75		0	0.0000	0.0000
			1	0.0000	0.0000
			2	0.0000	0.0000
			3	0.0004	0.0004
			4	0.0024	0.0028
			5	0.0115	0.0143
			6	0.0401	0.0544
			7	0.1032	0.1576
			8	0.1936	0.3512
			9	0.2581	0.6093
			10	0.2323	0.8416
			11	0.1267	0.9683
			12	0.0317	1.0000

i.e when C= 9. Which means. in 12 samples, if the number of satisfactory items is less than or equal to 9 the foreman should stop the machine