Exercises 1–3: Confidence Intervals

1. Exercise 1: Compute an 80% one-sided upper bound on lambda (a ceiling on the failure rate) — n = 200 products, failure-censored after 13 failures, lambda-hat = 0.0015 failures per hour.
2. Exercise 2: Compute a 90% two-sided confidence interval — n = 300 products, time-censored at 500 hours, r = 25 failures, lambda-hat = 0.01 failures per hour.
3. Exercise 3: Results analyzed under an exponential assumption give lambda-hat = 200 FITs, Type I censoring, 3 failures — compute a 95% two-sided CI for lambda. What if it was Type II censored? A FIT is one failure per billion (10⁹) device-hours, so 200 FITs = 200 × 10^-9 = 2 × 10^-7 failures per hour. The k-factor just multiplies lambda-hat, so you can work in FITs and convert at the end; either way, say which unit your answer is in.

A failure rate only hurts you when it is too high, so the question is a one-directional one: “how bad could lambda plausibly be?” That calls for a ceiling, a one-sided upper bound. A two-sided interval would spend half its error on a low tail nobody is worried about.

• Put all of alpha in the upper tail: an 80% one-sided upper bound uses the 80% column of the one-sided k-factor table (Table 3), not the 90% column a two-sided 80% interval would need.
• Exercise 1 is failure-censored (Type II), so look up r − 1 in Table 3 and multiply k by (r − 1)/r, as in the k-factor rules (Table 2).
• An upper bound on lambda is a lower bound on MTTF: take the reciprocal and the side flips.
• Always name the side. “An 80% one-sided bound” is incomplete until you say upper or lower.

The same idea, with the z version and a worked example, is in the chapter’s One-Sided Bounds subsection: R · Python.

Worked live in recitation, then answered as a pop quiz that opens on Canvas during class (1 pt) — graded on accuracy. It appears in this week’s module on Canvas once your instructor opens it.