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: Notation Find the values of \({\bf{\bar \bar x}}\)and\({\bf{\bar R}}\). Also find the values of LCL and UCL for an R chart, then find the values of LCL and UCL for an \({\bf{\bar x}}\) chart

The weights in grams are known for quarters minted by U.S. government.

Samples are taken each hour five times for 20 days.

Thus, each sequence of sample has size 5 (n) .

The averages for the sample mean and the range for the means are denoted as\(\bar \bar x,\;and\;\bar R\)respectively.

The mean of sample means:

\(\begin{array}{c}\bar \bar x = \frac{{\sum {\bar x} }}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{5.6334 + 5.6972 + ... + 5.8170}}{{20}}\\ = 5.6955\;g\end{array}\)

The mean of ranges:

\(\begin{array}{c}\bar R = \frac{{0.155 + 0.186 + ... + 0.602}}{{20}}\\ = 0.2054\end{array}\)

The values of\({D_3}\;{\rm{and}}\;{D_4}\)are obtained from the control charts constants table for sample size 5.

The lower and upper control limits are obtained as follows:

\(\begin{array}{c}L.C.L = {D_3} \times \bar R\\ = 0 \times 0.2054\\ = 0.0000\;{\rm{g}}\end{array}\)

\(\begin{array}{c}U.C.L = {D_4} \times \bar R\\ = 2.114 \times 0.2054\\ = 0.4342\;{\rm{g}}\end{array}\)

The L.C.L. and U.C.L. for R-chart is 0.0000 g and 0.4342 g respectively.

The value of\({A_2}\)is 0.577 which is obtained from the control charts constants table for sample size 5.

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. and U.C.L. for the \(\bar x - {\rm{chart}}\) is 5.5770 g and 5.8140 g respectively.