In many industrial production processes, measurements are made periodically on critical characteristics to ensure that the process is operating properly. Observations vary from item to item being produced, perhaps reflecting variability in material used in the process and/or variability in the way a person operates machinery used in the process. There is usually a target mean for the observations, which represents the long-run mean of the observations when the process is operating properly. There is also a target standard deviation for how observations should vary around that mean if the process is operating properly. A control chart is a method for plotting data collected over time to monitor whether the process is operating within the limits of expected variation. A control chart that plots sample means over time is called an x - chart. As shown in the following, the horizontal axis is the time scale and the vertical axis shows possible sample mean values. The horizontal line in the middle of the chart shows the target for the true mean. The upper and lower lines are called the upper control limit and lower control limit, denoted by UCL and LCL. These are usually drawn 3 standard deviations above and below the target value. The region between the LCL and UCL contains the values that theory predicts for the sample mean when the process is in control. When a sample mean falls above the UCL or below the LCL, it indicates that something may have gone wrong in the production process.

a. Walter Shewhart invented this method in 1924 at Bell Labs. He suggested using 3 standard deviations in setting the UCL and LCL to achieve a balance between having the chart fail to diagnose a problem and having it indicate a problem when none actually existed. If the process is working properly (€œin statistical control€) and if n is large enough that x has approximately a normal distribution, what is the probability that it indicates a problem when none exists? (That is, what€™s the probability a sample mean will be at least 3 standard deviations from the target, when that target is the true mean?)
b. What would the probability of falsely indicating a problem be if we used 2 standard deviations instead for the UCL and LCL?
c. When about nine sample means in a row fall on the same side of the target for the mean in a control chart, this is an indication of a potential problem, such as a shift up or a shift down in the true mean relative to the target value. If the process is actually in control and has a normal distribution around that mean, what is the probability that the next nine sample means in a row would (i) all fall above the mean and (ii) all fall above or all fall below the mean? (Use the binomial distribution, treating the successive observations as independent.)

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T-chart UCL Target LCL Time