During a production run of an injectable agent, 20 ampules...

During a production run of an injectable agent, 20 ampules are randomly sampled. Listed below are the volumes contained in each ampule. What are the tolerance limits, by volume, within which we would expect to find $99 \%$ of the total ampules in the run and have $99 \%$ confidence in our decision? $$ \begin{array}{llll} 1.99 & 2.00 & 2.02 & 1.98 \\ 2.01 & 2.01 & 2.01 & 2.02 \\ 2.00 & 1.98 & 1.99 & 2.00 \\ 1.98 & 1.99 & 2.00 & 2.01 \\ 2.03 & 2.00 & 2.00 & 1.99 \end{array} $$

During a production run of an injectable agent, 20 ampules are randomly sampled. Listed below are the volumes contained in each ampule. What are the tolerance limits, by volume, within which we would expect to find $99 \%$ of the total ampules in the run and have $99 \%$ confidence in our decision?
$$
\begin{array}{llll}
1.99 & 2.00 & 2.02 & 1.98 \\
2.01 & 2.01 & 2.01 & 2.02 \\
2.00 & 1.98 & 1.99 & 2.00 \\
1.98 & 1.99 & 2.00 & 2.01 \\
2.03 & 2.00 & 2.00 & 1.99
\end{array}
$$

Key Concepts

Tolerance Intervals

A tolerance interval is a statistical range that is constructed so as to contain a specified proportion of the population, with a certain level of confidence. Unlike a confidence interval for the mean, which estimates a parameter of the population, a tolerance interval aims to capture most of the individual values from the entire process. In this context, it is used to determine the volume limits that would cover 99% of all ampules produced.

Confidence Level

The confidence level reflects how sure we are that the tolerance interval actually covers the specified proportion of the population. For example, a 99% confidence level means that there is a 99% chance that the interval calculated from the sample data will contain the true percentage (here, 99%) of the production process values.

Sample Statistics

Sample statistics, including the sample mean and standard deviation, provide estimates of the central tendency and spread of the production process. These estimates are crucial for constructing tolerance intervals, especially under assumptions about the underlying distribution (commonly assumed to be normal in process control).

Statistical Process Control

Statistical process control involves monitoring and controlling a process through the use of statistical methods. Setting tolerance limits is an essential practice in quality assurance, ensuring that the process stays within acceptable bounds and that a predetermined percentage of products (in this case, ampules) meet the required specifications.

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