Report 1

Analysis of the Manufacturing Process

The current manufacturing process has been running for 23 hours each day for a total of three days. After the machine was running without interruptions, it was determined that the population mean (\(\mu\)) was 25, while the standard deviation (\(\sigma\)) was 1.75. Since the manufacturing process ran for three days without interruption, there are some concerns that the process has been taken out of control. To test this, samples (n) of 5 have been taken out to see if the manufacturing process should have been stopped.

Key Statistics

\(\mu=25\), \(\sigma=1.75\), \(n=5\)

To find the standard deviation of the sample means (\(\sigma_{\bar{x}}\)), the following computation was made:

\[\sigma_{\bar{x}}=\sigma/\sqrt{n}\] \[\sigma_{\bar{x}}=1.75/\sqrt{5} = 0.7826238\] The following computations were created to find the upper (UCL) and lower (LCL) control limits: \[UCL=\mu_\bar{x}+3\sigma_{\bar{x}}\] \[LCL=\mu_\bar{x}-3\sigma_{\bar{x}}\] \[UCL=25+3(0.7826238)=27.34787138\] \[LCL=25-3(0.7826238)=22.65212862\]

Day 1 Samples

The control chart below represents the samples collected on the first day of the manufacturing process:

The blue lines represent \(\mu\pm\sigma_{\bar{x}}\) and the red lines represent \(\mu\pm2\sigma_{\bar{x}}\)

According to the third Nelson rule, the manufacturing process should have stopped on the tenth hour because the four out of five consecutive points are above the center line and above \(\mu+\sigma_{\bar{x}}\), meaning that the process has gone out of control.

Day 2 Samples

This control chart represents the samples collected on the second day of the manufacturing process:

On the second day of the manufacturing process, the process should have stopped on the fourth hour since two out of three points consecutive points are above the center line and above \(\mu+2\sigma_{\bar{x}}\), breaking the second Nelson rule.

Day 3 Samples

This control chart represents the samples collected on the third day of the manufacturing process:

Since the manufacturing process did not break any of the Nelson Rules on the third day, it can be concluded that the process is in control.

Conclusion

On the first day of the manufacturing process, it fell out of control due to the third Nelson rule breaking. The second day was out of control due to the second Nelson rule breaking. The third day in the manufacturing process did not break any of the Nelson rules, meaning that the process is in control.