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?

The data for weights aregiven in the table.

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.

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.

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;

1. Draw a horizontal axis for days.
2. Draw a vertical axis for ranges.
3. Plot the data points for the range values corresponding to the day.
4. 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.

Two values lie above the upper control limit. It implies that the process is not stable as few measures are extreme.