Control Charts for p. In Exercises 5–12, use the given process data to construct a control chart for p. In each case, use the three out-of-control criteria listed near the beginning of this section and determine whether the process is within statistical control. If it is not, identify which of the three out-of-control criteria apply

Car Batteries Defective car batteries are a nuisance because they can strand and inconvenience drivers, and drivers could be put in danger. A car battery is considered to be defective if it fails before its warranty expires. Defects are identified when the batteries are returned under the warranty program. The Powerco Battery corporation manufactures car batteries in batches of 250, and the numbers of defects are listed below for each of 12 consecutive batches. Does the manufacturing process require correction?

3 4 2 5 3 6 8 9 12 14 17 20

Data are given on the number of defective batteries in 12 samples.

The sample size of each of the 12 samples is equal to 250.

Let\(\bar p\)be the estimated proportion of defectivebatteriesin all the samples.

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{samples}}}}\\ = \frac{{3 + 4 + 2 + ..... + 20}}{{12\left( {250} \right)}}\\ = \frac{{103}}{{3000}}\\ = 0.034333\end{array}\)

The value of\(\bar q\)is computed as shown:

\(\begin{array}{c}\bar q = 1 - 0.034333\\ = 0.965667\end{array}\)

The value of the lower control limit (LCL) is computed below:

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.034333 - 3\sqrt {\frac{{\left( {0.034333} \right)\left( {0.965667} \right)}}{{250}}} \\ = - 0.00021\\ \approx 0\end{array}\)

The value of the upper control limit (UCL) is computed below:

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.034333 + 3\sqrt {\frac{{\left( {0.034333} \right)\left( {0.965667} \right)}}{{250}}} \\ = 0.06888133\end{array}\)

The sample fraction defective for the ith batch can be computed as:

\({p_i} = \frac{{{d_i}}}{{250}}\)

Where,

\({p_i}\)be the sample fraction defective for the ith batch;

\({d_i}\)be the number of defective orders in the ith batch.

The computation of fraction defective for the ith batch is given as follows:

Follow the given steps to construct the p chart:

• Mark the values 1, 2, ...,12 on the horizontal axis and label the axis as “Sample”.
• Mark the values 0, 0.01, 0.02, ……, 0.09 on the vertical axis and label the axis as “Proportion”.
• Plot a straight line corresponding to the value “0.034333” on the vertical axis and label the line (on the left side) as “\(CL = 0.034333\)”.
• Plot a horizontal 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 “0. 0.0688813” on the vertical axis and label the line as “UCL=0.0688813.”
• Mark the given12 sample points (fraction defective of the ith lot) on the graph and join the dots using straight lines.

The following p chart is plotted:

The following criteria can be distinctly observed from the chart:

There appears to be an upward trend.

There is at least one point that lies beyond the upper control limit.

Since the above observations imply that the given process is not statistically stable, it can be concluded that the process is not within statistical control.

The proportion of defects in batteries is increasing. In order to produce good quality batteries, there is a need to take a corrective procedure to ensure that the number of defects declines.