ular interval can be computed if the sample means have a distribution that is close to nor- mal Acceptance intervals can also be computed for nonnormal probability distributions. Acceptance intervals find wide application for monitoring manufacturing processes to determine if product standards continue to be achieved. For example, in a manufactur- ing process the manufacturing engineer carefully sets and tests a new process so that it will produce products that all meet the guaranteed specifications for size, weight, or other measured properties . Thus the mean and standard deviation for the units produced are specified so that the desired product quality will be obtained. In addition, these inter- vals are also used for monitoring various business activities that involve customer service, Acceptance standards are established that meet stated marketing goals and customer ser vice-level capability. These standards in turn, are used to develop means, variances, and acceptance intervals to be used for process monitoring (Deming, 1986) However, it is possible that the process could come out of adjustment and produce detective product items. Changes in either the mean or variance of the critical measure ment result from a process that is out of adjustment Therefore, the process is monitored regularly by obtaining random samples and measuring the important properties, such as the sample mean and variance. If the measured values are within the acceptance interval then the process is allowed to continue If the values are not, then the process is stopped and necessary adjustments are made Acceptance intervals based on the normal distribution are defined by the distribution mean and variance. From the central limit theorvm we know that the sampling distribu- tion of sample means is often approximately normal, and thus acceptance intervals based on the normal distribution have wide applications Ansuming that we know the popula tion means and variance, then we can construct a symmetric acceptance interval provided that has a normal distribution and is the standard normal when the upper tall probability is a 2 The probability that the sample mean is included in the interval As noted, acceptance intervals are widely used for quality control monitoring of vari our production and service processes. The interval is ! 62. plotted over time the result is called an X-bar chart) and provides limits for the sample mean I given that the population mean is a typically, as very small (01), and standard practice in US industries is to use = 3. This is the source for the term Six Sigma used for various quality assurance programs (Hiam, 1942). If the sample mean is outside the acceptance interval, then we suspect that the population mean is not in a typical project engineers will take various steps to achieve a small variance for important prod- lict measurements that are directly related to product quality. Once the process has been adjusted so that the variance is small an acceptance interval for a sample mean-called a control internalis established in the form of a control chart (Montgomery, 1997). Then periodic random samples are obtained and compared to the control interval If the sample mean is within the control interval, it is concluded that the process is operating properly and no action is taken But if the sample mean is outside the control interval , it is concluded that the process is not operating properly and steps are taken to correct the process Example 6.5 Monitoring Health Insurance Claims (Acceptance Interval) Charlotte King, vice president of financial underwriting for a large health insurance company, wishes to monitor daily insurance claim payments to determine if the aver: age dollar value of subscriber claims is stable, increasing, or decreasing The value of individual claims varies up and down from one day to the next, and it would be naive to draw conclusions or change operations based on these daily variations. But at some point the changes become substantial and should be noted. She has asked you to de velop a procedure for monitoring the dollar value of individual claims