(a) The number of decays of a radioactive sample in 1 minute interval can be modeled by a Poisson distribution with ?. In five one-minute intervals, the number of radioactive decays that a Geiger counter recorded are
79 77 65 68 83
Find an estimation and an approximate 95% confidence interval for the number of decays per minutes.
(b) In a semiconductor manufacturing process, 100 chips are fabricated on each wafer.
From the quality inspection, the number of functional chips on 30 wafers are
64 61 63 62 64 55 52 56 58 65 63 59 55 58 60 55 57 47 57 56 66 57 63 63 68
58 58 56 60 53
Find an estimation of ?, the proportion of functional chips on a wafer, and an approximate 95% confidence interval for ?.
A quality characteristic is monitored by a control chart. After an out-of-control condition appears,the probability that this condition is detected by the first sample is 1−β. Suppose that the quality characteristics of the samples are independent, the out-of-control condition is persistent, and for each sample, the detection rate is the same. Find the following quantities:
(a) The probability that the out-of-control condition will be detected on the nth sample following the shift.
(b) The expected number of subgroups until the shift is detected.
(c) The probability that the first sample following the shift produces a statistic that plots inside the control limits.
(d) The probability that after the process shifts, ? consecutive samples produce statistics that plots outside of the control limits.
(e) The probability that at least 1 out of ? consecutive samples produce statistics that plots outside of the control limits.
What are the control limits for an x-bar control charts with α=0.004 probability limit and a sample size of 4? Assume that ? is the process standard deviation, and the mean shift is also ?. What is the type II error under this mean shift? Problem 4
Two decision rules are given here. Assume that the quality characteristic follows a normal distribution and the control chart has two-sigma control limits, and the sample size is ? = 4.
Rule 1: If one or more of the next 3 samples yield values of the sample average that fall outside the control limits, conclude that the process is out of control.
Rule 2: If all of the next 5 samples fall on the same side of the center line, conclude that the process is out of control.
(a) What is the Type-I error probability for each of these rules?
(b) If the mean of the quality characteristic shifts one process standard deviation, and remains there later, what is the Type-II error probability associated with each decision rule?
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