Final answer:
The CUSUM statistic is used for change detection by cumulatively summing deviations from the target mean. To plot the CUSUM, you generate random samples, calculate deviations from the mean, cumulatively sum these deviations, and create a chart. The CUSUM plot indicates shifts from the target mean over time.
Step-by-step explanation:
The cumulative sum (CUSUM) statistic is a sequential analysis technique used primarily for monitoring change detection. It is a form of a quality control method designed to detect a shift in the mean level of the measured variable from a known target value.
To derive the CUSUM statistic, you would typically start with a target mean (μ) and a standard deviation (σ). The CUSUM is calculated by taking the cumulative sum of the deviations of the sample means from the target mean, which can be positive or negative shifts.
To plot the CUSUM statistic for a sequence of randomly generated samples, follow these steps:
- Generate random samples assuming a known mean (μ) and standard deviation (σ).
- For each sample, calculate the sample mean.
- For each sample mean, calculate the deviation from the target mean.
- Sum these deviations cumulatively.
- Plot the resulting CUSUM statistic on a chart, with samples on the x-axis and the CUSUM on the y-axis.
The shape of the CUSUM plot will depend on whether the process mean is shifting over time. A consistent upward or downward trend in the chart suggests a shift from the target mean.
It is also worth noting that the CUSUM statistic is sensitive to small shifts in the process compared to other methods such as Shewhart control charts.