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%.
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.
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.
To calculate the centerline, upper control limit (UCL), and lower control limit (LCL) for the p-chart, we will use the following formulas:
Centerline (CL) = p̄
UCL = p̄ + 3 * √(p̄ * (1 - p̄)/n)
LCL = p̄ - 3 * √(p̄ * (1 - p̄)/n)
Given information:
p̄ (average percent of defective items) = 5.7% = 0.057
n (sample size) = 2800
a. Let's calculate the centerline, UCL, and LCL.
Centerline (CL) = p̄ = 0.057
UCL = p̄ + 3 * √(p̄ * (1 - p̄)/n)
UCL = 0.057 + 3 * √(0.057 * (1 - 0.057)/2800) ≈ 0.097
LCL = p̄ - 3 * √(p̄ * (1 - p̄)/n)
LCL = 0.057 - 3 * √(0.057 * (1 - 0.057)/2800) ≈ 0.017
b. The sample proportion of defective steel rods is 6.5% = 0.065, which lies between the UCL (0.097) and the centerline (0.057). Therefore, the correct answer is:
No, because the sample proportion lies between the upper and lower control limits.
To calculate the centerline (CL), upper control limit (UCL), and lower control limit (LCL) for the p̅ chart, we need to use the following formulas:
CL = p̅ = p = 5.7% = 0.057
UCL = p̅ + 3√(p̅(1-p̅)/n), where n is the batch size
LCL = p̅ - 3√(p̅(1-p̅)/n)
a. Let's plug in the values and calculate:
CL = p = 0.057
UCL = p̅ + 3√(p̅(1-p̅)/n) = 0.057 + 3√(0.057(1-0.057)/2800) ≈ 0.057 + 0.045 = 0.102
LCL = p̅ - 3√(p̅(1-p̅)/n) = 0.057 - 3√(0.057(1-0.057)/2800) ≈ 0.057 - 0.045 = 0.012
Therefore, the centerline, upper control limit, and lower control limit for the p̅ chart are approximately 0.057, 0.102, and 0.012, respectively.
b. The engineer inspects the next batch and finds that 6.5% are defective. To determine if the manufacturing process is under control, we need to check if the sample proportion falls within the control limits.
The sample proportion is 6.5%, which is greater than the upper control limit of 0.102. Therefore, the manufacturing process is NOT under control.
The correct choice is: No, because the sample proportion lies above the upper control limit.