Chapter 14: Q11BSC (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: \(\bar x\)-Chart Treat the 5 measurements from each day as a sample and construct an \(\bar x\)- chart. What does the result suggest?
Short Answer
The \(\bar x{\rm{ - chart}}\) is shown below:
The process appears to be out of control as a value lies above the upper control limit.
Step by step solution
Given information
The data for weights of quarters is taken for five different hours (n) for 20 days.
The table gives the corresponding data for the weights recorded for quality analysis of the process.
Step 2:Compute the average of mean measures and range for each day.
The two notations\(\bar \bar x,\bar R\)are the mean of all sample means for the samples and the mean of ranges for the samples respectively.
The mean of sample means and the ranges is computed as follows:
\(\begin{array}{c}\bar \bar x = \frac{{\sum {\bar x} }}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{5.6334 + 5.6972 + ... + 5.8170}}{{20}}\\ = 5.6955\;{\rm{g}}\end{array}\)
\(\begin{array}{c}\bar R = \frac{{{\rm{Sum}}\;{\rm{of}}\;{\rm{ranges}}}}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{0.155 + 0.186 + ... + 0.602}}{{20}}\\ = 0.2054\end{array}\)
Compute lower and upper control limit for \(\bar X\) - chart
The formulae for lower and upper control limits for the\(\bar x\)charts are:
\(\begin{array}{l}L.C.L. = \bar \bar x - \left( {{A_2} \times \bar R} \right)\\U.C.L. = \bar \bar x + \left( {{A_2} \times \bar R} \right)\end{array}\)
Where, the values of\({A_2}\)are taken from the table named as control charts constants table for the corresponding sample size measure.
For the sample size of 5 measurements;\({A_2} = 0.577\).
The lower and upper control limits are obtained as follows:
\(\begin{array}{c}L.C.L = \bar \bar x - \left( {{A_2} \times \bar R} \right)\\ = 5.6955 - 0.577 \times 0.2054\\ = 5.5770\;{\rm{g}}\end{array}\)
\(\begin{array}{c}U.C.L = \bar \bar x + \left( {{A_2} \times \bar R} \right)\\ = 5.6955 + \left( {0.577 \times 0.2054} \right)\\ = 5.8140\;{\rm{g}}\end{array}\)
Thus, the L.C.L. for the \(\bar x{\rm{ - chart}}\) is 5.5770 g and the U.C.L. for the \(\bar x{\rm{ - chart}}\)is 5.8140 g.
Sketch the\(\bar X\) - chart
The chart includes a centerline, lower and upper control limit which are:
\(\begin{array}{c}{\rm{Centerline}}\left( {\bar \bar x} \right) = 5.6955\;{\rm{g}}\\L.C.L. = 5.5770\;{\rm{g}}\\U.C.L. = 5.8140\,{\rm{g}}\end{array}\)
To sketch the graph;
- Draw a horizontal axis for days.
- Draw a vertical axis for sample means.
- Plot the data points for the sample mean values corresponding to the day.
- Mark the centerline\(\left( {\bar \bar x} \right)\) at 5.6955 g and upper and lower control limits as 5.5770 g and 5.8140 g respectively parallel to the horizontal axis.
Analyze the results
Using the\(\bar x\)-chart the sample means are analyzed.
As a value of thesample mean lies above the upper control limit line, it can be inferred that the process appears to be out-of-control from the statistical point of view.
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