A manufacturing process produces steel rods in batches of 2,800. The firm believes that the percent of defective items generated by this process is 5.7%.

Question

a. Calculate the centerline, the upper control limit (UCL), and the lower control limit (LCL) for the p¯
chart. (Round your answers to 3 decimal places.)





b. An engineer inspects the next batch of 2,800 steel rods and finds that 6.5% are defective. Is the manufacturing process under control?



multiple choice
Yes, because the sample proportion lies within the control limits.
Yes, because the sample proportion lies below the lower control limit.
No, because the sample proportion lies between the upper and lower control limits.
No, because the sample proportion lies below the lower control limit.

Answer 1

a. To calculate the centerline (CL) for the p-chart, we use the formula:

CL = p

where p is the expected proportion of defective items, given as 5.7%.

CL = 0.057

The upper control limit (UCL) for the p-chart is given by:

UCL = CL + (3 * √((CL/n) * (1-CL)))

where n is the sample size for each batch, given as 2,800.

UCL = 0.057 + (3 * √((0.057/2800) * (1-0.057)))
= 0.057 + (3 * √(0.00001279642857142857))
≈ 0.057 + (3 * 0.00357611458476118)
≈ 0.057 + 0.01072834375428353
≈ 0.067

The lower control limit (LCL) for the p-chart is given by:

LCL = CL - (3 * √((CL/n) * (1-CL)))

LCL = 0.057 - (3 * √((0.057/2800) * (1-0.057)))
= 0.057 - (3 * √(0.00001279642857142857))
≈ 0.057 - (3 * 0.00357611458476118)
≈ 0.057 - 0.01072834375428353
≈ 0.046

Therefore, the centerline (CL) is 0.057, the upper control limit (UCL) is 0.067, and the lower control limit (LCL) is 0.046.

b. The engineer inspects the next batch and finds that 6.5% of the steel rods are defective.

Since 6.5% (0.065) is within the control limits (0.046 to 0.067), we can conclude that the manufacturing process is under control.

Therefore, the answer is: Yes, because the sample proportion lies within the control limits.