The process standard deviation is estimated to be 0.12 ounces. Suppose that a beer bottler is filling containers labeled as 12 oz. Increasing the sensitivity of the control chart (reducing D) or decreasing the probability of a Type II error both result in a larger required sample size. This leaves us to trade off the chart sensitivity (D), the Type II error (b), and the required sample size (n). Furthermore, the process standard deviation (s), is typically estimated from the production process (rather than specified). Because for x-bar charts, the control limits are traditionally placed at ☓ standard deviations from the process average, the Type I error (a) is typically fixed at 0.0027. Selecting a sample size involves a trade-off between the above factors. As the change we are trying to detect is decreased, the sample size required to detect a process change increases (provided all other factors are unchanged). In other words, D represents the change in the process average that we are seeking to detect with the control chart.
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