please show all work :) Bonus Problem: Problem #4 (10 Bonus Points): With 30 sam

Question

please show all work :)

Bonus Problem: Problem #4 (10 Bonus Points): With 30 samples of constant sample size of 9, the following control chart centerline and limits for the X control chart have been developed for the sample (i.e, subgroup) size of 9 to be used in the monitoring stage: Upper Control Limit: 12 10 Lower Control Limit: 8 X Chart: Centerline: Part (a): (5 Points) What are the centerline and the upper control limit of an X chart for sample size of 16. Part (b): (5 Points) Suppose that the process had been in control but, at the beginning of and throughput a sampling period, the process mean ? ofX has shifted to 9 and the process standard deviation ? remains the same. What is the probability that this change of process mean will be detected by this X control chart of sample size 9 in this period?

Explanation / Answer

(a) Center line will not change and it would be same as CL = 10

Here UCL - CL = 3?/?n = 12 - 10 = 2

? = 2

so when n = 16 then

UCL = 10 + 3?/?n = 10 + 3* 2/?16 = 10 + 6/4 = 11.5

LCL = 10 - 3?/?n = 10 - 3* 2/?16 = 10 - 6/4 = 8.5

(b) Here process meaen ? of X has been shifted to 9

so, we will fail to reject when the process mean will be failed to detect.

Pr(12 < x < 8 ; 9 ; 2/3) = Pr(x < 12 ; 9 ; 2/3) - Pr(x < 8 ; 9 ; 2/3)

Z2 = (12 - 9)/(2/3) = 4.5 ; Z1 = (8 - 9)/(2/3) = -3/2 = -1.5

Pr(12 < x < 8 ; 9 ; 2/3) = Pr(x < 12 ; 9 ; 2/3) - Pr(x < 8 ; 9 ; 2/3) = Pr(Z < 4.5) - Pr(Z < -1.5)

= 0.999997 - 0.066807

= 0.9332

so probability that this change in process mean will be detected will be (1 - 0.9332) = 0.0668

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