Chapter 14: Q10BSC (page 654)
Quarters. In Exercises 9–12, refer to the accompanying table of weights (grams) of quarters minted by the U.S. government. This table is available for download at www.TriolaStats.com.
Day | Hour 1 | Hour 2 | Hour 3 | Hour 4 | Hour 5 | \(\bar x\) | s | Range |
1 | 5.543 | 5.698 | 5.605 | 5.653 | 5.668 | 5.6334 | 0.0607 | 0.155 |
2 | 5.585 | 5.692 | 5.771 | 5.718 | 5.72 | 5.6972 | 0.0689 | 0.186 |
3 | 5.752 | 5.636 | 5.66 | 5.68 | 5.565 | 5.6586 | 0.0679 | 0.187 |
4 | 5.697 | 5.613 | 5.575 | 5.615 | 5.646 | 5.6292 | 0.0455 | 0.122 |
5 | 5.63 | 5.77 | 5.713 | 5.649 | 5.65 | 5.6824 | 0.0581 | 0.14 |
6 | 5.807 | 5.647 | 5.756 | 5.677 | 5.761 | 5.7296 | 0.0657 | 0.16 |
7 | 5.686 | 5.691 | 5.715 | 5.748 | 5.688 | 5.7056 | 0.0264 | 0.062 |
8 | 5.681 | 5.699 | 5.767 | 5.736 | 5.752 | 5.727 | 0.0361 | 0.086 |
9 | 5.552 | 5.659 | 5.77 | 5.594 | 5.607 | 5.6364 | 0.0839 | 0.218 |
10 | 5.818 | 5.655 | 5.66 | 5.662 | 5.7 | 5.699 | 0.0689 | 0.163 |
11 | 5.693 | 5.692 | 5.625 | 5.75 | 5.757 | 5.7034 | 0.0535 | 0.132 |
12 | 5.637 | 5.628 | 5.646 | 5.667 | 5.603 | 5.6362 | 0.0235 | 0.064 |
13 | 5.634 | 5.778 | 5.638 | 5.689 | 5.702 | 5.6882 | 0.0586 | 0.144 |
14 | 5.664 | 5.655 | 5.727 | 5.637 | 5.667 | 5.67 | 0.0339 | 0.09 |
15 | 5.664 | 5.695 | 5.677 | 5.689 | 5.757 | 5.6964 | 0.0359 | 0.093 |
16 | 5.707 | 5.89 | 5.598 | 5.724 | 5.635 | 5.7108 | 0.1127 | 0.292 |
17 | 5.697 | 5.593 | 5.78 | 5.745 | 5.47 | 5.657 | 0.126 | 0.31 |
18 | 6.002 | 5.898 | 5.669 | 5.957 | 5.583 | 5.8218 | 0.185 | 0.419 |
19 | 6.017 | 5.613 | 5.596 | 5.534 | 5.795 | 5.711 | 0.1968 | 0.483 |
20 | 5.671 | 6.223 | 5.621 | 5.783 | 5.787 | 5.817 | 0.238 | 0.602 |
Quarters: R Chart Treat the five measurements from each day as a sample and construct an R chart. What does the result suggest?
Short Answer
The R-chart is shown below:
The process appears to be out of control as two values lie above the upper control limit.
Step by step solution
Given information
The data for weights aregiven in the table.
Step 2:Compute the average of range measures
The notation\(\bar R\)is the mean of all range values in the table.
The formula for the mean of range is computed as follows:
\(\begin{array}{c}\bar R = \frac{{{\rm{Sum}}\;{\rm{of}}\;{\rm{ranges}}}}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{0.155 + 0.186 + ... + 0.602}}{{20}}\\ = 0.2054\end{array}\)
Thus, the mean of range measures is 0.2054 g.
Compute lower and upper control limit for R - chart
The formulae for lower and upper control limits for R-chart are:
\(\begin{array}{l}L.C.L. = {D_3} \times \bar R\\U.C.L. = {D_4} \times \bar R\end{array}\)
The values for the measures are computed corresponding to sample size 5 from the control chart constants table as 0 and 2.114 respectively.
The lower and upper control limits are obtained as follows:
\(\begin{array}{c}L.C.L. = 0 \times 0.2054\\ = 0.0000\;{\rm{g}}\end{array}\)
\(\begin{array}{c}U.C.L = 2.114 \times 0.2054\\ = 0.4342\;{\rm{g}}\end{array}\)
Thus, the LCL for the R-chart is 0.0000 g and the UCL is 0.4342 g.
Sketch the R– chart
The chart includes a centerline, lower and upper control limit which are:
\(\begin{array}{c}{\rm{Centerline}}\left( {\bar R} \right) = 0.2054\;{\rm{g}}\\L.C.L. = 0.0000\;{\rm{g}}\\U.C.L. = 0.4342\,{\rm{g}}\end{array}\)
To sketch the graph;
- Draw a horizontal axis for days.
- Draw a vertical axis for ranges.
- Plot the data points for the range values corresponding to the day.
- Mark the centerline\(\left( {\bar R} \right)\) at 0.2054 g and upper and lower control limits as 0.4342 g and 0.000 g respectively parallel to the horizontal axis.
Analyze the results
Two values lie above the upper control limit. It implies that the process is not stable as few measures are extreme.
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