The Probability of a Confidence Interval Based on Minimal Estimates of the Mean and the Standard Deviation

A confidence interval is an interval in which a measurement falls with a given probability [1]. This paper addresses the question of defining the probability of a confidence interval that can be constructed when only a minimal number of measurements are known. The problem has a couple of significant applications. In geophysics, time-lapse 3D seismic monitoring is used to monitor oil and gas reservoirs before and after production. Because of the high cost of 3D seismic monitoring, a minimal-effort time-lapse 3D seismic monitoring approach was proposed by Houston and Kinsland [2]. This approach proposes a minimal measurement interval based on preliminary measurements. It implies successively smaller seismic surveys asmonitoring continues and knowledge of the behavior of the reservoir grows. The success of the minimal-effort method is based on the probability that the reservoir is detected by a seismic survey. The minimal-effort method is based on a minimal number of measurements, which connects it to the topic of this paper. Another significant application of a confidence interval based on a minimal number of measurements is the problem of cancer treatment based on the irradiation of a tumor [3]. The purpose of irradiation is to destroy the tumor, but targeting the tumor can be problematic because tumors often exhibit small random motions within the body. Because of random tumor motion, the result of radiation treatment often includes the destruction of healthy tissue. Clearly, it is beneficial that cancer radiation treatment incorporates minimal intervals. The success of the cancer treatment is based on the probability that the tumor is irradiated while using a minimal irradiation interval. This success hinges on the ability of the radiation treatment to incorporate preliminary measurements that are often minimal. It is upon this basis that the cancer treatment problem connects to the topic of this paper. In this paper, we construct a confidence interval based on minimal estimates of the mean and the standard deviation. Explicitly, we use two measurements to specify an interval that contains a subsequent measurement with a given probability. The mathematical effort includes the derivation of a specific probability for the confidence interval. The probability is computed using the cumulative distribution function, and this probability is averaged over both the estimate of the mean and the sample standard deviation to yield a specific value.The results are compared to a computer simulation that estimates the probability based on frequency.