At a major supermarket, checkout time is monitored to improve service quality. For six days, samples each with 20 observations have been obtained, and the following table summarizes the data. Determine if the checkout process is in control or not. You must show your analysis for the support of your decision.
Sample Mean Range
1 3.06 .42
2 3.15 .50
3 3.11 .41
4 3.13 .46
5 3.06 .46
6 3.09 .45
To determine if the checkout process is in control or not, we need to perform a control chart analysis. Control charts are commonly used in process improvement to monitor the stability and predictability of a process over time.
In this case, we can use the X-bar chart and the Range chart to analyze the data. The X-bar chart helps us analyze the process mean, while the Range chart helps us analyze the process variability.
Let's start with the X-bar chart:
1. Calculate the overall average of the sample means:
- Add up the mean values from all six samples: 3.06 + 3.15 + 3.11 + 3.13 + 3.06 + 3.09 = 18.60.
- Divide the sum by the number of samples: 18.60 / 6 = 3.10.
2. Calculate the average range:
- Add up the range values from all six samples: 0.42 + 0.50 + 0.41 + 0.46 + 0.46 + 0.45 = 2.70.
- Divide the sum by the number of samples: 2.70 / 6 = 0.45.
3. Calculate the control limits for the X-bar chart:
- Upper Control Limit (UCL) = X-bar + A2 * Range
- Lower Control Limit (LCL) = X-bar - A2 * Range
- A2 is a constant based on the sample size, and for n = 20, A2 = 0.729.
- UCL = 3.10 + 0.729 * 0.45 = 3.43
- LCL = 3.10 - 0.729 * 0.45 = 2.77
4. Plot the sample means on the X-bar chart:
- Plot each sample mean value on a graph, keeping track of the sample number on the x-axis and the means on the y-axis.
- Draw the control limits (UCL and LCL) as horizontal lines on the chart.
Next, let's move on to the Range chart:
1. Plot the sample ranges on the Range chart:
- Plot each sample range value on a graph, using the same axis conventions as the X-bar chart.
- Calculate the UCL and LCL for the Range chart using the common formula:
- UCL = D4 * Range
- LCL = D3 * Range
- For n = 20, D4 = 2.282 and D3 = 0.
2. Draw the control limits (UCL and LCL) as horizontal lines on the chart.
Finally, we need to analyze the control charts and make a decision:
- X-bar chart: If all the sample means fall within the control limits (UCL and LCL) and show no patterns or trends, the process is considered in control. If any sample means fall outside the control limits or show patterns/trends, the process is considered out of control.
- Range chart: If all the sample ranges fall within the control limits (UCL and LCL) and show no patterns or trends, the process is considered in control. If any sample ranges fall outside the control limits or show patterns/trends, the process is considered out of control.
Carefully examine both charts to see if any points fall outside the control limits or exhibit patterns/trends. Based on the results, you can determine if the checkout process is in control or not.
Note: In case of any outliers or patterns, further investigation might be needed to identify the root cause of the issue and take appropriate actions for process improvement.