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Other Numerical Examples of Confidence Intervals and Bounds for the Exponential Case

We now construct a 80% CI for the mean of the n = 45 data given in Table 1.

Chi-Square Percentile (90 DF, α/2 = 0.1) = 73.2911
Chi-Square Percentile (90 DF, 1 - α/2 = 0.9) = 107.5650

Constructing a 80% CI for the failure rate, λ:

Constructing a 80% CI for reliability (T = 100):

R1(T) = Exp{-T/θ1} = Exp{-λ1T} =
Exp (-0.008*100) = 0.442674

R2(T) = Exp{-T/θ2) = Exp{-λ2T} =
Exp (-0.012*100) = 0.302375

Proceeding likewise, we obtain the CI limits for the mean, rates and reliability for mission times of T = 50, 30, and 20 hours as shown in Table 2. For comparison, the exact reliability values for lives that are distributed Exponential (100) are also given.

Table 2. CIs for Mean, Rate, and Reliability
Parameter Upper Lim for 80% CI Lower Lim for 80% CI Exact Reliability
Mean 122.711 83.6060 N/A
Rate 0.0119609 0.0081492 N/A
Reliability
for T =
100 0.442674 0.302375 0.367879
50 0.665337 0.549887 0.606531
30 0.783114 0.698496 0.740818
20 0.849604 0.787244 0.818731