A reliability test was terminated after 171 hours based upon a pre-established plan. An estimate of mean-time-to-failure of 57 hours was obtained. The lower 95 percent confidence bound on the true…

Question

Options

• A125 hours.
• B22 hours.
• C27 hours.
• D11 hours.

• A
  9% (5)
• B
  81% (43)
• C
  4% (2)
• D
  6% (3)

Explanation

Option B (22 hours) is correct because this is a time-terminated (Type I censored) reliability test, which requires the chi-square lower confidence bound formula: Lower MTTF = 2T / χ²(α, 2r+2). With test time T = 171 hours and estimated MTTF = 57 hours, we get r = 171/57 = 3 failures. Plugging in: 2(171) / χ²(0.05, 8) = 342 / 15.507 ≈ 22 hours.

Why the distractors fail:

• C (27 hours) is the trap - it uses χ²(0.05, 6) (degrees of freedom 2r instead of 2r+2), which is the formula for a failure-terminated test, not a time-terminated one.
• A (125 hours) is far too high for a lower bound on a 57-hour MTTF estimate; it would exceed the estimate itself, which is impossible for a lower bound.
• D (11 hours) likely results from misapplying a higher confidence level (e.g., 99%) or wrong degrees of freedom.

Memory tip: For time-terminated tests, add 2 extra degrees of freedom (2r+2) because the test end-time itself carries information - think "Time-terminated = Two extra."

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