Question

**Please answer entirety of question in all of its parts!
Thank you!**

It is known from Phase I data, the process has the average defects rate of 40 defects per 1000 boards.

(a) One engineer suggests a sampling strategy to make quality checking every hour by selecting 100 boards, how to construct a monitoring chart (CL = ? UCL = ? LCL = ?)?

(b) How many boards should be checked at each sampling hour in order to obtain a positive lower control limit?

(c) One month later, this engineer decides to change the sampling strategy by checking 400 boards at each sampling hour. Under this new sampling strategy, how do you suggest for continuously monitoring such a process (selecting a c chart or u chart?). Please define the random variable that follows the Poisson distribution, and provide the distribution parameter correspondingly.

(d) For (c), if a u chart is suggested, what is a convenient unit that you like to suggest? What is the sample size? What are the corresponding control limits of this u chart (CL, UCL, and LCL)? What are c0 and u0 when the process is in-control?

(e) If a c chart is still used for (c), what are the control limits (CL, UCL, and LCL)?

(f) If at time t, it is found that a sample of 400 boards has 22 defects, what are the monitoring statistics to be plotted in above charts of (d) and (e)?

Answer #1

**SOLUTION**:

p = no of defects proportion = 0.04 **a)** 100 board
to be selected. For 100 board **mean = n *p = 4.**
Hence CL =4 and It varies both sides from higher to lower.
**Variance = n* p* q = 0.24** std dev =
0.49 If confidence interval 95% is taken, then
**LCL = 4-0.29(0.49)/10 = 3986** **UCL =
4.0142.**

**b)**
If the margin of error is fixed as 0.005 say 0.005 =
2.58(0.49)/rtn then no of boards to be checked = 25.84^2 = 63928
Sample size >63928

**C) ** For 400 charts we first draw a
chart using 99% confidence with mean n*p = 16

Then check whether this lies within this range every hour
continuously.

**d.) ** if only u chart is
suggested then size should be double of 63928

that is = 127856.

**f) **If at a time t,it is found that a
sample of 400 boards has 22 defects then mean = 22.

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