A textbook publisher wants to determine process control charts for the textbook height. The following table shows the first 20 samples of n=4 sample size of the production of the text.

Sample - X Bar - Range
1 - 10.01 - .01
2 - 9.97 - .04
3 - 9.98 - .03
4 - 10.01 - .04
5 - 10.00 - .04
6 - 10.01 - .02
7 - 10.04 - .04
8 - 9.99 - .07
9 - 9.95 - .04
10 - 9.99 - .06
11 - 10.02 - .03
12 - 10.01 - .04
13 - 10.04 - .02
14 - 9.95 - .02
15 - 10.04 - .01
16 - 9.99 - .04
17 - 10.04 - .03
18 - 10.04 - .06
19 - 10.04 - .07
20 - 10.02 - .06

What are the upper and lower control limits for the X bar chart using the data from the samples above?

What are the upper and lower control limits for the R chart using the data from the samples above?

Discuss the charts you have plotted. What conclusions and recommendations do you make from those charts and the data?

To determine the upper and lower control limits for the X bar chart, we need to calculate the average (X bar) of the samples and calculate the standard deviation (s) of the sample means. The control limits can then be calculated using the following formulas:

Upper Control Limit (UCL) = X double bar + A2 * R bar
Lower Control Limit (LCL) = X double bar - A2 * R bar

Where X double bar is the average of the sample means, R bar is the average of the ranges, and A2 is a constant value based on the sample size.

To calculate the upper and lower control limits for the R chart, we need to calculate the control limits for the ranges, using the following formulas:

Upper Control Limit (UCL) = D4 * R bar
Lower Control Limit (LCL) = D3 * R bar

Where R bar is the average of the ranges, D4 and D3 are constant values based on the sample size.

Let's calculate these values:

First, let's calculate the X bar values:
X Bar = (10.01 + 9.97 + 9.98 + 10.01) / 4 = 9.9925
X Bar = (10.01 + 10 + 10.01 + 10.04) / 4 = 10.015
X Bar = (9.99 + 9.95 + 9.99 + 10.02) / 4 = 9.9875
...
X Bar = (10.04 + 10.04 + 10.02 + 10.02) / 4 = 10.03

Next, let's calculate the R values:
Range = 0.01
Range = 0.04
Range = 0.03
...
Range = 0.06

Now, let's calculate the averages for X bar and R:
Average X Bar (X double bar) = (9.9925 + 10.015 + 9.9875 + ... + 10.03) / 20 = 10.006
Average R (R bar) = (0.01 + 0.04 + 0.03 + ... + 0.06) / 20 = 0.036

For a sample size of 4, the constants A2, D3, and D4 are as follows:

A2 = 0.729
D3 = 0
D4 = 2.282

Now, let's calculate the control limits:

Upper Control Limit (X bar chart) = 10.006 + 0.729 * 0.036 = 10.032
Lower Control Limit (X bar chart) = 10.006 - 0.729 * 0.036 = 9.98

Upper Control Limit (R chart) = 2.282 * 0.036 = 0.082
Lower Control Limit (R chart) = 0 * 0.036 = 0

Now, let's discuss the charts and draw conclusions from the data:

The X bar chart is used to monitor the average (X bar) of a process over time. In this case, the X bar chart shows that the average textbook height is centered around 10.006, with upper and lower control limits at 10.032 and 9.98, respectively. From the X bar chart, we can conclude that the process appears to be in control, as none of the sample means fall outside the control limits.

The R chart is used to monitor the variability and range (R) of the process over time. In this case, the R chart shows that the range of textbook heights varies around 0.036, with upper and lower control limits at 0.082 and 0, respectively. From the R chart, we can see that there is some variation in the textbook heights, but it is within acceptable limits.

Based on the charts and data, we can conclude that the textbook height process is in control and is producing textbooks with a consistent average height and acceptable variation. No significant outliers or patterns are observed in the charts, suggesting a stable and predictable process. However, it is always recommended to monitor the process over time to ensure continued control and make adjustments if necessary.