Question

Energy Consumption. Exercises 1–5 refer to the amounts of energy consumed in the author’s home. (Most of the data are real, but some are fabricated.) Each value represents energy consumed (kWh) in a two-month period. Let each subgroup consist of the six amounts within the same year. Data are available for download atwww.TriolaStats.com.

	Jan.-Feb.	Mar.-April	May-June	July-Aug.	Sept.-Oct.	Nov.-dec.
Year 1	3637	2888	2359	3704	3432	2446
Year 2	4463	2482	2762	2288	2423	2483
Year 3	3375	2661	2073	2579	2858	2296
Year 4	2812	2433	2266	3128	3286	2749
Year 5	3427	578	3792	3348	2937	2774
Year 6	4016	3458	3395	4249	4003	3118
Year 7	5261	2946	3063	5081	2919	3360
Year 8	3853	3174	3370	4480	3710	3327

Energy Consumption: Notation After finding the values of the mean and range for each year, find the values of\(\bar \bar x\)and\(\bar R\). Then find the values of LCL and UCL for an R chart and find the values of LCL and UCL for an\(\bar x\)chart.

Short answer

The following table shows the mean and range of each of the eight samples:

							Sample Mean \(\left( {{{\bar x}_i}} \right)\)	Sample Range \(\left( {{R_i}} \right)\)
Sample 1	3637	2888	2359	3704	3432	2446	3077.667	1345
Sample 2	4463	2482	2762	2288	2423	2483	2816.833	2175
Sample 3	3375	2661	2073	2579	2858	2296	2640.333	1302
Sample 4	2812	2433	2266	3128	3286	2749	2779	1020
Sample 5	3427	578	3792	3348	2937	2774	2809.333	3214
Sample 6	4016	3458	3395	4249	4003	3118	3706.5	1131
Sample 7	5261	2946	3063	5081	2919	3360	3771.667	2342
Sample 8	3853	3174	3370	4480	3710	3327	3652.333	1306

The value of\(\bar \bar x\)is equal to 3157 kWh.

The value of\(\bar R\)is equal to 1729 kWh.

The values of LCL and UCL for the R chart are 0 kWh and 3465 kWh,respectively.

The values of LCL and UCL for the \(\bar x\) chart are 2322 kWh and 3992 kWh,respectively.

Step-by-step solution

Given information

Data values are given for eight years on the energy consumed (in kWh) in a two-month period.

The sample size for each of the eight years is equal to 6.

Step 2:Mean of each sample

Let n denote the sample size of each sample.

Here,\(n = 6\).

The total number of samples is denoted by N.

Here,\(N = 8\).

The mean of sample 1 is computed below:

\(\begin{aligned}{c}{{\bar x}_1} = \frac{{3637 + 2888 + 2359 + 3704 + 3432 + 2446}}{6}\\ = 3077.67\end{aligned}\)

The mean of sample 2 is computed below:

\(\begin{aligned}{c}{{\bar x}_2} = \frac{{4463 + 2482 + 2762 + 2288 + 2423 + 2483}}{6}\\ = 2816.83\end{aligned}\)

The mean of sample 3 is computed below:

\(\begin{aligned}{c}{{\bar x}_3} = \frac{{3375 + 2661 + 2073 + 2579 + 2858 + 2296}}{6}\\ = 2640.33\end{aligned}\)

The mean of sample 4 is computed below:

\(\begin{aligned}{c}{{\bar x}_4} = \frac{{2812 + 2433 + 2266 + 3128 + 3286 + 2749}}{6}\\ = 2779.0\end{aligned}\)

The mean of sample 5 is computed below:

\(\begin{aligned}{c}{{\bar x}_5} = \frac{{3427 + 578 + 3792 + 3348 + 2937 + 2774}}{6}\\ = 2809.33\end{aligned}\)

The mean of sample 6 is computed below:

\(\begin{aligned}{c}{{\bar x}_6} = \frac{{4016 + 3458 + 3395 + 4249 + 4003 + 3118}}{6}\\ = 3706.5\end{aligned}\)

The mean of sample 7 is computed below:

\(\begin{aligned}{c}{{\bar x}_7} = \frac{{5261 + 2946 + 3063 + 5081 + 2919 + 3360}}{6}\\ = 3771.67\end{aligned}\)

The mean of sample 8 is computed below:

\(\begin{aligned}{c}{{\bar x}_8} = \frac{{3853 + 3174 + 3370 + 4480 + 3710 + 3327}}{6}\\ = 3652.33\end{aligned}\)

Range of each sample

The range of sample 1 is computed below:

\(\begin{aligned}{c}Rang{e_1} = Ma{x_1} - Mi{n_1}\\ = 3704 - 2359\\ = 1345.0\end{aligned}\)

The range of sample 2 is computed below:

\(\begin{aligned}{c}Rang{e_2} = Ma{x_2} - Mi{n_2}\\ = 4463 - 2288\\ = 2175.0\end{aligned}\)

The range of sample 3 is computed below:

\(\begin{aligned}{c}Rang{e_3} = Ma{x_3} - Mi{n_3}\\ = 3375 - 2073\\ = 1302.0\end{aligned}\)

The range of sample 4 is computed below:

\(\begin{aligned}{c}Rang{e_4} = Ma{x_4} - Mi{n_4}\\ = 3286 - 2266\\ = 1020.0\end{aligned}\)

The range of sample 5 is computed below:

\(\begin{aligned}{c}Rang{e_5} = Ma{x_5} - Mi{n_5}\\ = 3792 - 578\\ = 3214.0\end{aligned}\)

The range of sample 6 is computed below:

\(\begin{aligned}{c}Rang{e_6} = Ma{x_6} - Mi{n_6}\\ = 4249 - 3118\\ = 1131.0\end{aligned}\)

The range of sample 7 is computed below:

\(\begin{aligned}{c}Rang{e_7} = Ma{x_7} - Mi{n_7}\\ = 5261 - 2919\\ = 2342.0\end{aligned}\)

The range of sample 8 is computed below:

\(\begin{aligned}{c}Rang{e_8} = Ma{x_8} - Mi{n_8}\\ = 4480 - 3174\\ = 1306.0\end{aligned}\)

The following table summarizes the sample means and sample ranges:

							Sample Mean \(\left( {{{\bar x}_i}} \right)\)	Sample Range \(\left( {{R_i}} \right)\)
Sample 1	3637	2888	2359	3704	3432	2446	3077.667	1345
Sample 2	4463	2482	2762	2288	2423	2483	2816.833	2175
Sample 3	3375	2661	2073	2579	2858	2296	2640.333	1302
Sample 4	2812	2433	2266	3128	3286	2749	2779	1020
Sample 5	3427	578	3792	3348	2937	2774	2809.333	3214
Sample 6	4016	3458	3395	4249	4003	3118	3706.5	1131
Sample 7	5261	2946	3063	5081	2919	3360	3771.667	2342
Sample 8	3853	3174	3370	4480	3710	3327	3652.333	1306

Value of \(\bar \bar x\)

The value of\(\bar \bar x\)is computed below:

\(\begin{aligned}{c}\bar \bar x = \frac{{{{\bar x}_1} + {{\bar x}_2} + {{\bar x}_3} + {{\bar x}_4} + {{\bar x}_5} + {{\bar x}_6} + {{\bar x}_7} + {{\bar x}_8}}}{8}\\ = \frac{{3077.67 + 2816.83 + 2640.33 + 2779 + 2809.33 + 3706.5 + 3771.67 + 3652.33}}{8}\\ = 3156.708\\ \approx 3157\end{aligned}\)

Thus, the value of \(\bar \bar x\) is equal to 3157 kWh.

Value of \(\bar R\)

The value of\(\bar R\)is computed below:

\(\begin{aligned}{c}\bar R = \frac{{{R_1} + {R_2} + {R_3} + {R_4} + {R_5} + {R_6} + {R_7} + {R_8}}}{8}\\ = \frac{{1345 + 2175 + 1302 + 1020 + 3214 + 1131 + 2342 + 1306}}{8}\\ = 1729.375\\ \approx 1729\end{aligned}\)

Thus, the value of \(\bar R\) is equal to 1729 kWh

Limits of R chart

The lower control limit (LCL) is computed using the given formula:

\(LCL = {D_3}\bar R\)

The value of the constant\({D_3}\)for n equal to 6 is equal to 0.000.

\(\begin{aligned}{c}LCL = {D_3}\bar R\\ = 0.000\left( {1729} \right)\\ = 0\end{aligned}\)

Thus, LCL is equal to 0 kWh.

The upper control limit (UCL) is computed using the given formula:

\(UCL = {D_4}\bar R\)

The value of the constant\({D_4}\)for n = 6 is equal to 2.004.

\(\begin{aligned}{c}UCL = {D_4}\bar R\\ = 2.004\left( {1729} \right)\\ = 3464.916\\ \approx 3465\end{aligned}\)

Thus, UCL is equal to 3465 kWh.

Limits of \(\bar x\) chart

The lower control limit (LCL) is computed using the given formula:

\(LCL = \bar \bar x - {A_2}\bar R\)

The value of the constant\({A_2}\)for n equal to 6 is equal to 0.483.

\(\begin{aligned}{c}LCL = \bar \bar x - {A_2}\bar R\\ = 3157 - \left( {0.483} \right)\left( {1729} \right)\\ = 2321.893\\ \approx 2322\end{aligned}\)

Thus, LCL is equal to 2322 kWh.

The upper control limit (UCL) is computed using the given formula:

\(UCL = \bar \bar x + {A_2}\bar R\)

The value of the constant\({A_2}\)for n = 6 is equal to 0.483.

\(\begin{aligned}{c}UCL = \bar \bar x + {A_2}\bar R\\ = 3157 + \left( {0.483} \right)\left( {1729} \right)\\ = 3992.107\\ \approx 3992\end{aligned}\)

Thus, UCL is equal to 3992 kWh.