Question

A manufacturing process is in-control and centered. A critical quality characteristic is normally distributed with a mean of 20 and a standard deviation of 2. The DPMO of the process is 318.  

(1) What is the upper specification limit for the characteristic?

(2) The daily production rate is 1000 parts. How many parts per day would you expect to have a dimension less than 21 but greater the 19.5?

(3) A 3-sigma Xbar chart based on a sample of size 4 is monitoring the process. What is the probability that the chart signals if the process mean shifts to 24?

Answer 1

1) for DPMO =318 ; p value in tail area =318/10⁶ =0.000318 ; for which corresponding z =3.6

hence upper specification limit for the characteristic =mean+z*std deviation =20+3.6*2=27.2

2)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	20
std deviation   =σ=	2.0000

probability =

P(19.5<X<21)

=

P(-0.25<Z<0.5)=

0.6915-0.4013=

0.2902

expected parts dimension less than 21 but greater the 19.5 =np=1000*0.2902=290

3)probability that the chart signals if the process mean shifts to 24 =P(Xbar>26)=P(Z>(26-24)*sqrt(4)/2)=P(Z>2)=0.0228