Final answer:
To calculate the control limits for the process with a mean of 55.0 units and population standard deviation of 1.72 units, we find the standard error and apply it to the formula for 3-sigma control limits. The Upper Control Limit (UCL) is 56.29 units and the Lower Control Limit (LCL) is 53.71 units.
Step-by-step explanation:
To determine the Upper Control Limit (UCL) and Lower Control Limit (LCL) for the process with a mean of 55.0 units, a population standard deviation of 1.72 units, and a sample size of 16, we use the formula for control limits in statistical process control.
The Upper Control Limit (UCL) is 56.29 units and the Lower Control Limit (LCL) is 53.71 units.
The general formula for the Upper Control Limit (UCL) and Lower Control Limit (LCL) on an x-bar chart is given by:
UCL = process mean + z* (standard deviation/√n)
LCL = process mean - z* (standard deviation/√n)
Where process mean is the mean of the process, standard deviation is the population standard deviation, n is the sample size, and z is the z-value for the desired confidence level.
For a 3-sigma control chart, z is equal to 3 since it is the multiple of standard deviations from the mean (3 standard deviations covers approximately 99.73% of the data for a normal distribution).
First, we calculate the standard error, which is the standard deviation divided by the square root of the sample size (1.72/√16 = 1.72/4 = 0.43).
Then, we can calculate UCL and LCL as follows:
UCL = 55.0 + 3 * 0.43 = 55.0 + 1.29 = 56.29 units
LCL = 55.0 - 3 * 0.43 = 55.0 - 1.29 = 53.71 units
Therefore, the Upper Control Limit is 56.29 units and the Lower Control Limit is 53.71 units.