This is a NAVAL POSTGRADUATE SCHOOL MONTEREY CA report procured by the Pentagon and made available for public release. It has been reproduced in the best form available to the Pentagon. It is not spiral-bound, but rather assembled with Velobinding in a soft, white linen cover. The Storming Media report number is A178483. The abstract provided by the Pentagon follows: Estimating the failure time of a product with a high degree of confidence is a difficult endeavor. Clearly, if the product is inexpensive and fails quickly, extensive tests can be run to make prediction more accurate. When the item under scrutiny is expensive, not prone to failure, or both, calculating accurate estimates and confidence bounds (CBs) becomes more difficult. Furthermore, many methods currently in use are prone to error, sometimes making a critical part appear more reliable than it actually is. Much of our military uses end-items that fall into this category. The lives of our soldiers, sailors, airmen, and Marines often depend on accurate reliability estimates for the equipment and weapons they work on every day. This thesis first introduces reliability and the common techniques for measuring it. Secondly, it shows that these estimates are often biased. Next, this bias is quantified using Monte Carlo simulation and corrected through simple tables and equations. The tables and equations can be used to map nominal confidence bounds to actual confidence bounds. Lastly, these results are applied to a Marine Corps program and a test run at a major automotive brake system manufacturer. These examples will illustrate the impact of uncorrected bias and what can be done to correct it.