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	4016	3458	3395	4249	4003	3118
Year 8	4016	3458	3395	4249	4003	3118

Energy Consumption: R Chart Let each subgroup consist of the 6 values within a year. Construct an R chart and determine whether the process variation is within statistical control. If it is not, identify which of the three out-of-control criteria lead to rejection of statistically stable variation

Short answer

The following is the constructed R chart for the given samples:

Since none of the three out-of-control criteria are met, the constructed R chart depicts the process variation within statistical control.

Step-by-step solution

Given information

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

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

Important lines in R Chart

For constructing the R chart, the values of the central line$\overline{R}$, the lower control limit (LCL), and the upper control limit (UCL) need to be determined.

Referring to Exercise 1 CRE, the values are as follows:

$\begin{array}{r}l\overline{R}=1729\mathrm{k}\mathrm{W}\mathrm{h}\\ LCL=0\mathrm{k}\mathrm{W}\mathrm{h}\\ UCL=3465\mathrm{k}\mathrm{W}\mathrm{h}\end{array}$

Construction of R chart

Follow the given steps to construct the R chart:

• Mark the values 1, 2, ..., 8 on the horizontal axis and label the axis as “Sample.”
• Mark the values 0, 500, 1000, …., 3500 on the vertical axis and label the axis as “Sample Range.”
• Plot a straight line corresponding to the value “1729” on the vertical axis and label the line (on the left side) as “$\overline{R}$=1729.”
• Plot ahorizontal line corresponding to the value “0” on the vertical axis and label the line as “LCL=0.”
• Similarly, plot a horizontal line corresponding to the value “3465” on the vertical axis and label the line as “UCL=3465.”
• Mark the following sample ranges on the graph corresponding to the sample number and join the dots using straight lines:

Sample No.	Sample Ranges
1	1345
2	2175
3	1302
4	1020
5	3214
6	1131
7	2342
8	1306

The following R chart is plotted:

Analysis of the R chart

The chart does not seem to show any of the three out-of-control criteria.

Thus, it can be concluded that the process variation is within statistical control.