Final answer:
A process is in control when the sample points are randomly distributed with an equal number above and below the center line, close to the center line without trending, and within control limits. The central limit theorem supports quality control by ensuring the normal distribution of means for sufficiently large sample sizes, aiding the interpretation of control processes.
Step-by-step explanation:
A process is generally considered to be in control when the sample points are randomly distributed with about the same number above the center line as below, most of the sample points are close to the center line without trending toward the control limits, and there are no sample points outside of the control limits. These characteristics suggest that the process is stable and predictable, adhering to statistical control principles.
The central limit theorem (CLT) is very relevant in this context, as it assures that if samples of sufficient size are drawn from a population (the size is often considered sufficient if n ≥ 30), then the distribution of the sample means will tend to be normal, or bell-shaped, regardless of the shape of the population distribution. Therefore, when monitoring a process using control charts, it is essential the sample size is sufficient to ensure that the sampling distribution of the mean follows a normal distribution, which in turn simplifies the interpretation of the control chart.
In the context of control processes, the central limit theorem allows quality control professionals to use the properties of the normal distribution to make inferences and decisions about the process. For example, if the means of the samples from the process are within the control limits and show no patterns or trends suggesting systemic variations, the process is considered to be in a state of statistical control.