146 Reliability of a “one-shot” device. 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 .
Note: The U.S. Department of Defense Reliability Analysis Center (DoD RAC) provides designers with free access to tables and toolboxes that give the minimum sample size n required to obtain a desired confidence level for a specified number of observed failures in the sample.