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?

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.

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:

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

\(\begin{array}{c}\bar R = \frac{{{\rm{Sum}}\;{\rm{of}}\;{\rm{ranges}}}}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}}}\\ = \frac{{0.155 + 0.186 + ... + 0.602}}{{20}}\\ = 0.2054\end{array}\)

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.

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;

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

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.