I am confuse the how to calculate the sgima level of process, please tell me below which one is right and which is wrong clearly.

We have 1,000,000 productions, find defect is 8,000. defect opportunity is 1.

Then case 1
DPMO is 8,000 according to the ASQ six sigmal primer appendix, the sigmal level is 2.9 sigma(with 1.5 shift)

case2
DPU is 0.008 DPMO=0.008
Y= RTY= e-DPU=0.992
Probablity of defect is P(d)=1-Y = 1-0.992=0.008
then use Z value table is 2.41
and add 1.5 shift, then the sigma level is 2.41+1.5=3.9 sigmal

Case3.
using formula sigmal level= 0.8406+ SQRT(29.37-(2.221ln(PPM)))
then 0.846+SQRT(29.37-(2.221ln8000))=3.9 sigmal

question 1:
the resuult of case 2 and 3 are different with case1 which one is right?
question2:
if we want to know the cpk of this process, is 8,000/1,000,000=0.8% just P value(because it is attribute data), that is right?

It agrees with the second two calculations (i.e. 3.9 sigma). I have to assume the table you used in Case 1 is wrong. Perhaps it was for a shift of 0.5 instead of 1.5. Perhaps it was simply mis-typed.

Due to higher sources of variation, long term capability is alway lower than short term capability.
Six sigma always refers to short term capability. A factor of 1.5 sigma is applied to account for common cause meanshift which can happen both in favorable or unfavorable directions.

Six sigma capability in Short term is expected to reduce upto 6-1.5= 4.5 sigma in Long term. When you divide sigma by 3, you get capability. This means that short term six sigma or capability of 2.0 is expected to reduce to 4.5 sigma or capability of 1.50.

In your case, if you are interested in short term, sigma value is 3.9 which can reduce to 2.4 in long term.

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