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 at www.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:\(\bar x\)Chart Let each subgroup consist of the 6 values within a year. Construct an\(\bar x\)chart and determine whether the process mean is within statistical control. If it is not, identify which of the three out-of-control criteria lead to rejection of a statistically stable mean.

Short answer

The following is the constructed \(\bar x\) chart for the given samples:

There seems to be an upward shift in the sample means, which indicates that the process is not stable.

Thus, it can be said that the process mean is not 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 \(\bar x\) Chart

For constructing the\(\bar x\)chart, the values of the central line\(\bar \bar x\), 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:

\(\begin{aligned}{l}\bar \bar x = 3157\;{\rm{kWh}}\\LCL = 2322\;{\rm{kWh}}\\UCL = 3992\;{\rm{kWh}}\end{aligned}\)

Construction

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 2500, 3000, ……,4000 on the vertical axis and label the axis as “Sample Mean.”
• Plot a straight line corresponding to the value “3157” on the vertical axis and label the line (on the left side) as “\(\bar \bar x\)=3157.”
• Plot a horizontal line corresponding to the value “2322” on the vertical axis and label the line as “LCL=2322.”
• Similarly, plot a horizontal line corresponding to the value “3992” on the vertical axis and label the line as “UCL=3992.”
• Mark the sample means on the graph and join the dots using straight lines.

Sample No.	Sample Means
1	3078
2	2817
3	2640
4	2779
5	2809
6	3707
7	3772
8	3652

The following \(\bar x\)chart is plotted:

Analysis of the\(\bar x\)chart

Here, the initial few points lie very low in the chart, and the points at the end lie very high.

This indicates that there has been an upward shift in the values, which violates the stability/control of the process.

Thus, it can be concluded that the process mean is not within statistical control.