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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 鈥渋n control鈥� 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鈥檚 test, the mean and standard deviation are per square inch and 22.9 psi, respectively. i.e. $n=9$, $\stackrel{炉}{x}=985.6$, $s=22.9$.

Also, it is given that the process is to be considered 鈥渙ut of control鈥� if the true mean strength differs from 1000psi.

To check whether the process is 鈥渙ut of control鈥� 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:

$\stackrel{炉}{x}卤t_{伪/2,\left(n-1\right)}脳\frac{s}{\sqrt{n}}$

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

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

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

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

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.