A “one-shot” device can be used only once; after use, the device (e.g., a nuclear weapon, space shuttle, automobile air bag) either is destroyed or must be rebuilt. The destructive nature of a one-shot device makes repeated testing either impractical or too costly. Hence, the reliability of such a device must be determined with minimal testing. Consider a one-shot device that has some probability p of failure. Of course, the true value of p is unknown, so designers will specify a value of p which is the largest defective rate that they are willing to accept. Designers will conduct n tests of the device and determine the success or failure of each test. If the number of observed failures, x, is less than or equal to some specified value k, then the device is considered to have the desired failure rate. Consequently, the designers want to know the minimum sample size n needed so that observing K or fewer defectives in the sample will demonstrate that the true probability of failure for the one-shot device is no greater than p.
a. Suppose the desired failure rate for a one-shot device is p = .10. Suppose also that designers will conduct n = 20 tests of the device and conclude that the device is performing to specifications if K = 1 (i.e., if 1 or no failure is observed in the sample). Find P(x ≤ pbc).
b. In reliability analysis, 1 – P(x ≤ K) is often called the level of confidence for concluding that the true failure rate is less than or equal to p. Find the level of confidence for the one-shot device described in part a. In your opinion, is this an acceptable level? Explain.
c. Demonstrate that the confidence level can be increased by either (1) increasing the sample size n or (2) decreasing the number K of failures allowed in the sample.
d. Typically, designers want a confidence level of .90, .95, or .99. Find the values of n and K to use so that the designers can conclude with at least 95% confidence that the failure rate for the one-shot device of part a is no greater than p = .10.