1. Samples of ?? 4 items each are taken from a manufacturing process at regular intervals.Suppose the process mean ?? ? 200 and the standard deviation ?? 10.
10 a. Find both the 3 sigma control limits and the probability limits ( ? ? 0.002) to control the process mean.12 b. Find the average run length and the standard deviation of the run length for both charts (in a.), in the following cases: i) process is in control, ii) process mean shifts to ?? ? 209.
14 c. For both charts in a., what is the required sample size to detect this shift in the process mean on the first or second sample following the shift with the probability ? 0.9? For which of the two charts is the required sample size bigger and why? 14 d. Assume that the following rule is applied to the control chart with 3 sigma limits constructed in a. The process is declared out of control when either a point plots outside the 3 sigma limits or two consecutive points plot between the two and three sigma control limits on the same side of the control chart. If the process mean shifts to ?? ? 209, what is the probability that the shift will be detected on the first sample following the shift?. A production process was designed for mass production of ball bearings 0.5 in. in diameter. A quality control engineer randomly sampled four ball bearings every hour for 15 hours. The following data resulted: Hour Diameter of ball bearing (in.)
1 0.481 0.511 0.463 0.495
2 0.507 0.486 0.491 0.491
3 0.495 0.483 0.48 0.508
4 0.511 0.453 0.539 0.494
5 0.479 0.51 0.554 0.499
6 0.488 0.479 0.497 0.52
7 0.511 0.508 0.501 0.51
8 0.507 0.522 0.521 0.515
9 0.494 0.521 0.503 0.497
10 0.522 0.482 0.524 0.477
11 0.509 0.501 0.53 0.52
12 0.518 0.523 0.47 0.486
13 0.49 0.473 0.48 0.5
14 0.464 0.498 0.473 0.521
15 0.464 0.476 0.513 0.491

Assume that the specification limits are: LSL=0.42 in. and USL=0.58 in.

12 a. Check, if the process is in statistical control. Use ? and R charts with probability limits, ?? 0.002. If necessary, revise the trial control limits. Then, check normality by plotting the in-control data on the attached normal paper.12 b. Check the claim that ?? ? 1.3 at ?? 0.05 significance level and estimate the fractionnonconforming below the LSL and above the USL. Find the 95% confidence interval for ??. Isthe process centered at the target value? Compute ??? and ???.