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16-61. Suppose a process is in control, and 3-sigma control limits are in use on the X chart. Let the mean shift by 1.5 . What is the probability that this shift will remain undetected for three consecutive samples What would its probability be if 2-sigma control limits were used The sample size is 4. 16-62. Consider an X control chart with k-sigma control limits. Develop a general expression for the probability that a point will plot outside the control limits when the process mean has shifted by units from the center line. 16-63. Suppose that an X chart is used to control a normally distributed process and that samples of size n are taken every n hours and plotted on the chart, which has k-sigma limits. (a) Find a general expression for the expected number of samples and time that will be taken until a false action signal is generated. (b) Suppose that the process mean shifts to an out-ofcontrol state, say 1 . Find an expres0 sion for the expected number of samples that will be taken until a false action is generated.

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(c) Evaluate the in-control ARL for k 3. How, does this change if k 2 What do you think about the use of 2-sigma limits in practice (d) Evaluate the out-of-control ARL for a shift of 1 sigma, given that n 5. 16-64. Suppose a P chart with center line at p with ksigma control limits is used to control a process. There is a critical fraction defective pc that must be detected with probability 0.50 on the rst sample following the shift to this state. Derive a general formula for the sample size that should be used on this chart. 16-65. Suppose that a P chart with center line at p and k-sigma control limits is used to control a process. What is the smallest sample size that can be used on this control chart to ensure that the lower control limit is positive 16-66. A process is controlled by a P chart using samples of size 100. The center line on the chart is 0.05. What is the probability that the control chart detects a shift to 0.08 on the rst sample following the shift What is the probability that the shift is detected by at least the third sample following the shift 16-67. Consider a process where speci cations on a quality characteristic are 100 15. We know that the standard deviation of this normally distributed quality characteristic is 5. Where should we center the process to minimize the fraction defective produced Now suppose the mean shifts to 105 and we are using a sample size of 4 on an X chart. What is the probability that such a shift will be detected on the rst sample following the shift What is the average number of samples until an out-of-control point occurs Compare this result to the average number of observations until a defective occurs (assuming normality). 16-68. The NP Control Chart. An alternative to the control chart for fraction defective is a control chart based on the number of defectives, or the NP control chart. The chart has centerline at n p, and the control limits are UCL LCL np np 3 2np11 3 2np11 p2 p2 (b) Apply this control chart to the data in Example 16-4. (c) Will this chart always provide results that are equivalent to the usual P chart 16-69. The EWMA Control Chart. The exponentially weighted moving average (or EWMA) is de ned as follows:

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where 0 1, and the starting value of the EWMA at time t 0 is Z0 0 (the process target). An EWMA control chart is constructed by plotting the Zt values on a chart with center line at 0 and appropriate control limits. (a) Verify that E1Zt 2 0 (b) Let 2t be V1Zt 2 , and show that z

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