Samples of 2000 were taken from an encoding machine at a bank over a five-week period.Management is willing to use the historical average for the central line of the chart. Which one of the following would be the limits for an attribute control chart with z = 2? A)UCL 0.010 and LCL 0.001 B)UCL 0.010 and LCL

Question

Samples of 2000 were taken from an encoding machine at a bank over a five-week period.Management is willing to use the historical average for the central line of the chart. Which one of the following would be the limits for an attribute control chart with z = 2? A)UCL > 0.010 and LCL > 0.001 B)UCL > 0.010 and LCL < 0.001 C)UCL < 0.010 and LCL > 0.001 D)UCL < 0.010 and LCL < 0.001

Quizplus · Accepted Answer

Since we are dealing with an attribute control chart, we need to use the formula for the control limits: 
UCL = p + z * sqrt [(p(1-p))/n]
LCL = p - z * sqrt [(p(1-p))/n]
where p is the historical average, z is the number of standard deviations from the mean (given as 2), and n is the sample size (given as 2000).

Since management is willing to use the historical average for the central line, we don't need to calculate it. We just need to plug in the values: 
UCL = p + 2 * sqrt [(p(1-p))/2000]
LCL = p - 2 * sqrt [(p(1-p))/2000]

To find out whether the limits are above or below 0.001 and 0.010, we need to make some assumptions about the historical average. Since we don't have any information about it, we can assume that it's somewhere between 0.001 and 0.010 (since the encoding machine is presumably accurate).

If we assume that p is closer to 0.010, we can plug in p = 0.010 and see if the limits meet that requirement: 
UCL = 0.010 + 2 * sqrt [(0.010*(1-0.010))/2000] = 0.012
LCL = 0.010 - 2 * sqrt [(0.010*(1-0.010))/2000] = 0.008

Since both limits are above 0.010, this option is eliminated.

If we assume that p is closer to 0.001, we can plug in p = 0.001 and see if the limits meet that requirement: 
UCL = 0.001 + 2 * sqrt [(0.001*(1-0.001))/2000] = 0.00247
LCL = 0.001 - 2 * sqrt [(0.001*(1-0.001))/2000] = -0.00047

Since the LCL is negative, we can assume it's actually 0 (since we can't have negative defects). Therefore, the limits for an attribute control chart with z = 2 would be: 
UCL < 0.010 and LCL > 0.001

Samples of 2000 Were Taken from an Encoding Machine at a Bank