
Concept explainers
A new type of smoke detector battery is developed. From laboratory tests under standard conditions, the half-lives (defined as less than 50 percent of full charge) of 20 batteries are shown below. (a) Make a histogram of the data and/or a
(a)

Sketch a histogram and normal probability plot for the sample.
Explain whether the battery half-life can be assumed normal or not.
Answer to Problem 55CE
The histogram for the sample is,
The normal probability plot for the sample is,
The battery half-life can be assumed normal.
Explanation of Solution
Calculation:
The given information is that, the half-lives of the 20 batteries are considered.
Histogram for hours:
Software procedure:
Step-by-step procedure to obtain the histogram for ‘hours’ using the MINITAB software:
- • Choose Graph > Histogram.
- • Choose Simple, and then click OK.
- • In Graph variables, enter the corresponding column ‘Hours’.
- • Click OK
Normal Probability plot for hours:
Software procedure:
Step-by-step procedure to obtain the normal probability plot for ‘hours’ using the MINITAB software:
- • Choose Graph > Probability Plot.
- • Choose Single, and then click OK.
- • In Graph variables, enter the column of Hours.
- • Click OK.
Justification: From the histogram it can be observed that the hours’ is slightly bell-shaped and could be slightly normal. Also, from the normal probability plot it can be observed that the data points of ‘hours all lies within the lines forming the linear pattern and is approximately normal. Overall the distribution of the half-life battery is normally distributed.
Hence, battery half-life can be assumed normal.
(b)

Find the centerline and control limits for the
Answer to Problem 55CE
The centerline is 8,760 and control limits for an
Explanation of Solution
Calculation:
The given information is that, the subgroup size is
Control limits with known
If the values of
In the formula,
Control limits for an
Substitute,
Hence, the control limits for an
(c)

Find the centerline and control limits for the
Answer to Problem 55CE
The centerline is 8,784.8 and control limits for an
Explanation of Solution
Calculation:
The given information is that, the half-lives of the 20 batteries are considered. The subgroups size is
Software procedure:
Step-by-step procedure to obtain the sample mean and standard deviation for ‘hours’ using the MINITAB software:
- • Choose Stat > Basic Statistics > Display Descriptive Statistics.
- • In Variables enter the columns Hours.
- • Choose option statistics, and select Mean, Standard deviation.
- • Click OK.
Output using MINITAB software is,
The sample mean is 8,784.8 and sample standard deviation is 216.1.
Empirical control limits:
If the value of
In the formula,
Control limits for an
Substitute,
Hence, the control limits for an
(d)

Explain whether the control limits from the sample would be reliable or not.
Suggest an alternative.
Answer to Problem 55CE
The control limits from the sample would not be reliable.
The alternative approach is constructing R chart.
Explanation of Solution
Justification: The mean of battery half-life is
Alternative: Since the sample size is large for determining the capability of the process the range chart can be used as an alternative approach. The R chart is a control chart that shows the changes of the range value over a period of time and determines variation around the mean by using the sample ranges.
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Chapter 17 Solutions
APPLIED STAT.IN BUS.+ECONOMICS
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