Solved Problems

A structural steel fabrication plant monitors the length of beams. Over 20 samples of size $n=5$ were taken. The average of the sample means is $\bar{\bar{X}} = 1000$ mm, and the average range is $\bar{R} = 15$ mm. From SQC tables for $n=5$, $A_2 = 0.577$, $D_3 = 0$, and $D_4 = 2.114$. Determine the control limits for the $\bar{X}$ and $R$ charts.

An inspector counts the number of surface defects on precast concrete panels. Over 25 panels inspected, a total of 50 defects were found. Determine the control limits for a $c$-chart to monitor future panel production.

A civil engineering firm monitors the quality of asphalt batches from a supplier. They take 30 random samples of size $n = 100$ batches over a month. Across all 3000 batches tested, 120 are found to be defective (e.g., incorrect temperature or aggregate mix). Construct the control limits for a $p$-chart to monitor the fraction defective in future samples of size 100.

A manufacturing process produces structural bolts with a specified target diameter of $20.0$ mm and tolerance limits of $19.8$ mm to $20.2$ mm. A random sample of bolts is measured, revealing that the process mean diameter is $\bar{x} = 20.05$ mm with a standard deviation of $s = 0.04$ mm. Calculate the process capability ratio ($C_p$) and the process capability index ($C_{pk}$). Is the process capable of consistently meeting the specifications?