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MIL-HDBK-338 15 OCTOBER 1984 8.3.2.5.2 CONFIDENCE LIMITS - EXPONENTIAL DISTRIBUTION Two situations have to be considered for estimating confidence intervals: one in which the test is run until a preassigned number of failures (r*) occurs, and one in which the test is stopped after a preassigned number of test hours (t*) is accumulated. The formula for the confidence interval employs the x 2 (chi-square) distribution. A short table of x 2 values are given in Table 8.3.2.5.2-1. The general notation used is X2 (P, d) where p and d are two constants used to choose the correct value from the table. The quantity p is a function of the confidence coefficient; d, known as the degrees of freedom, is a function of the number of failures. Equations (8.12) and (8.13) are for one-sided or two-sided 100 (1-a) percent confidence intervals. For non-replacement tests with a fixed truncation time, the limits are only approximate. Tv/Jc- IT Equations for Confidence Limits on Mean Life Type of Confidence Limits One Sided (Lower Limit) Two Sided (Upper and Lower Limits) Fixed Number of failures, r* (fa,2r ') / 2T 2T \ (/(|,2r)' X2U-i,2r)J Fixed Truncationt Time t* ' ( ^ A ^Xs(a.2r+2)' ) / 2T 2T \ (x"2(!,2r+2)' XS(l-^2r)J tFor_jwn-replacement__tests^ only one-sided intervals are possible T5r r = Tr.~~TTse "Sri"degrees oFTreedom for the lower limit if T~= rr: " The terms used are identified as follows: n = number of items placed on test at time t = 0 t* = time at which the life test is terminated 8 = mean life (or MTBF for the case of replacement or repair upon failure) X^ . 2r+2), for example, is the _*_percentage point of the chi-square distribution for (2r+2) degrees of freedom r = number of failures accumulated at time t* r* = preassigned number of failures a = acceptable risk of error 1-a = confidence level 8-39

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