Twelve samples, each containing five parts, were taken
from a process that produces steel rods. The length of each rod in the
samples was determined. The results were tabulated and sample
means and ranges were computed. The results were:
A) Determine the upper and lower control limits and the overall
means for Ẋ charts and R - Charts.
B) Draw the charts and plot the values of the sample means and
ranges.
C) Do the data indicate a process that is in control?
D) Why or why not?
To answer these questions, we need to understand the concepts of control limits, central tendency, and variability, as well as how to calculate them for X-bar (sample means) and R-charts (sample ranges). Here is how you can compute these values and interpret the results:
A) To determine the upper and lower control limits and the overall means for X-bar and R-charts, you need to perform the following calculations:
1. Calculate the sample means (Ẋ) for each sample. Sum the lengths of each rod in a sample and divide by the number of rods in that sample (5 in this case). Repeat this process for all 12 samples.
2. Calculate the overall mean (Ẋ-double bar) by summing all the sample means obtained in step 1 and dividing by the number of samples (12 in this case).
3. Calculate the ranges (R) for each sample. Subtract the smallest length from the largest length in each sample. Repeat this process for all 12 samples.
4. Calculate the average range (R-bar) by summing all the ranges obtained in step 3 and dividing by the number of samples (12 in this case).
5. Calculate the upper and lower control limits for the X-bar chart using the formulas:
Upper Control Limit (UCL-Xbar) = Ẋ-double bar + A2 * R-bar
Lower Control Limit (LCL-Xbar) = Ẋ-double bar - A2 * R-bar
A2 is a constant value based on the sample size, which you can refer to in statistical tables.
6. Calculate the upper and lower control limits for the R-chart using the formulas:
Upper Control Limit (UCL-R) = D4 * R-bar
Lower Control Limit (LCL-R) = D3 * R-bar
D3 and D4 are constant values based on the sample size, which you can also refer to in statistical tables.
B) To draw the charts and plot the values of the sample means and ranges, you need to create a graph with the X-axis representing the sample number (1 to 12) and the Y-axis representing the length of the rods. Mark the upper and lower control limits for both the X-bar and R-charts on the graph. Then plot the sample means and ranges for each sample as points on the graph.
C) To determine if the process is in control, we need to examine whether the plotted points on both the X-bar and R-charts fall within the control limits. If most of the points fall within the control limits and there is no specific pattern or trend, then the process is considered to be in control.
D) The data indicate whether a process is in control or not based on the results obtained in step C. If most of the plotted points fall within the control limits and show no specific patterns or trends, then the process is likely in control. Otherwise, if there are points outside of the control limits or any patterns or trends observed, then the process may be out of control, indicating some issue or variation in the production of steel rods.
By following these steps, you should be able to compute the control limits, draw the control charts, and determine whether the process is in control based on the obtained data.