Question

• 6.32. Control charts for ¯xx¯ and R are in use with the following parameters:

  ¯¯¯xx¯ Chart	R Chart
  UCL = 363.0	UCL = 16.18
  Center line = 360.0	Center line = 8.91
  LCL = 357.0	LCL = 1.64

  The sample size is n = 9. Both charts exhibit control. The quality characteristic is normally distributed.

  1. What is the α-risk associated with the ¯xx¯ chart?
  2. Specifications on this quality characteristic are 358 ± 6. What are your conclusions regarding the ability of the process to produce items within specifications?
  3. Suppose the mean shifts to 357. What is the probability that the shift will not be detected on the first sample following the shift?
  4. What would be the appropriate control limits for the ¯xx¯ chart if the type I error probability were to be 0.01?
• 6.33. A normally distributed quality characteristic is monitored through use of an ¯xx¯ and an R chart. These charts have the following parameters (n = 4):

  ¯¯¯xx¯ Chart	R Chart
  UCL = 626.0	UCL = 18.795
  Center line = 620.0	Center line = 8.236
  LCL = 614.0	LCL = 0

  Both charts exhibit control.

  1. What is the estimated standard deviation of the process?
  2. Suppose an s chart were to be substituted for the R chart. What would be the appropriate parameters of the s chart?
  3. If specifications on the product were 610 ± 15, what would be your estimate of the process fraction nonconforming?
  4. What could be done to reduce this fraction nonconforming?
  5. What is the probability of detecting a shift in the process mean to 610 on the first sample following the shift (σ remains constant)?
  6. What is the probability of detecting the shift in part (e) by at least the third sample after the shift occurs?

6.35. The following ¯xx¯ and s charts based on n = 4 have shown statistical control:

¯¯¯xx¯ Chart	s Chart
UCL = 710	UCL = 18.08
Center line = 700	Center line = 7.979
LCL = 690	LCL = 0

1. Estimate the process parameters μ and σ.
2. If the specifications are at 705 ± 15, and the process output is normally distributed, estimate the fraction nonconforming.
3. For the ¯xx¯ chart, find the probability of a type I error, assuming σ is constant.
4. Suppose the process mean shifts to 693 and the standard deviation simultaneously shifts to 12. Find the probability of detecting this shift on the ¯xx¯ chart on the first subsequent sample.
5. For the shift of part (d), find the average run length.

Answer 1