Background:

Quality Improvement Partners $($QIP$)$ is a consulting firm that advises clients about sampling and statistical procedures that can be used to control their manufacturing processes. Your company has given QIP a sample of $800$ observations taken during a time in which your process was operating satisfactorily.

The design specification indicates the mean measurement for the process should be $12.$ The sample standard deviation for these data was $0\mathrm{.}21.$ With that much data, you can assume the population standard deviation is $0\mathrm{.}21.$

QIP has suggested that periodic random samples of size $30$ be taken so you can monitor the process on an ongoing basis. By analyzing the new samples, you can quickly learn whether the process is operating satisfactorily or if it is operating differently than the expected mean. When the process is not operating satisfactorily, corrective action can be taken to attempt to eliminate the root cause$($s$)$ of the errors $($assuming you know what they are$).$ You will take corrective action whenever you reject the null hypothesis at the $0\mathrm{.}01$ significance level. Use this information to answer the questions in this Data Lab.

Compute limits for the sample mean $\stackrel{}{x}$ around $=12$ such that, as long as a new sample mean is within those limits$,$ the process will be considered to be operating satisfactorily. If $\stackrel{}{x}$ exceeds the upper limit or if

$\stackrel{}{x}$ is below the lower limit$,$ corrective action will be taken. For quality control purposes, these limits are

referred to as lower and upper control limits$,$ respectively:

Lower Control Limit $=,($Round answers to $4$ decimal places$)$

Upper Control Limit $=|,($Round answers to $4$ decimal places$)$

$*$Be careful when you enter these values so that they are entered in the correct boxes.

Hint: Your alpha level will play a role in this!