“Out of control” production processes. When companies employ control charts to monitor the quality of their products, a series of small samples is typically used to determine if the process is “in control” during the period of time in which each sample is selected. (We cover quality-control charts in Chapter 13.) Suppose a concrete-block manufacturer samples nine blocks per hour and tests the breaking strength of each. During 1 hour’s test, the mean and standard deviation are 985.6 pounds per square inch (psi) and 22.9 psi, respectively. The process is to be considered “out of control” if the true mean strength differs from 1,000 psi. The manufacturer wants to be reasonably certain that the process is really out of control before shutting down the process and trying to determine the problem. What is your recommendation?

When companies employ control charts to monitor the quality of their products, a series of small samples is typically used to determine if the process is “in control” during the period of time in where each sample is selected. Suppose a concrete block manufacturer samples nine blocks per hour and tests the breaking strength of each. During one hour’s test, the mean and standard deviation are per square inch and 22.9 psi, respectively. i.e. $n=9$, $\overline{x}=985.6$, $s=22.9$.

Also, it is given that the process is to be considered “out of control” if the true mean strength differs from 1000psi.

To check whether the process is “out of control” or not, calculate 95% (assumed) confidence interval for the mean breaking strength.

The 95% confidence interval for the mean breaking strength is given as follows:

$\overline{x}\pm t_{\alpha /2,\left(n-1\right)}\times \frac{s}{\sqrt{n}}$

Where, $\overline{x}$ is the sample mean, s is sample standard deviation, n is the sample size and $t_{\alpha /2,\left(n-1\right)}$ is the critical value of t at 8 degrees of freedom.

Here, $t_{\alpha /2,\left(n-1\right)}=2.306$

Then, the 95% confidence interval for the mean breaking strength is calculated as follows:

$$\begin{gathered} 985.6\pm 2.306\times \frac{22.9}{\sqrt{9}} \\ 985.6\pm 17.602 \\ \left(967.998,1003.202\right) \end{gathered}$$

From the above discussion, it can be seen that the 95% confidence interval for mean breaking strength includes the true mean strength, 1000psi.

Hence, it may conclude that the mean breaking strength does not differ from 1000 psi.

Therefore, it can recommend to the manufacturer that process is under control and need not shutdown the process.