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

The sample size of each of the 12 samples is equal to 200.

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

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{samples}}}}\\ = \frac{{5 + 7 + 4 + ..... + 19}}{{12\left( {200} \right)}}\\ = \frac{{100}}{{2400}}\\ = 0.041667\end{array}\)

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

\(\begin{array}{c}\bar q = 1 - 0.041667\\ = 0.958333\end{array}\)

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

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.041667 - 3\sqrt {\frac{{\left( {0.041667} \right)\left( {0.958333} \right)}}{{200}}} \\ = - 0.000723\\ \approx 0\end{array}\)

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

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.041667 + 3\sqrt {\frac{{\left( {0.041667} \right)\left( {0.958333} \right)}}{{200}}} \\ = 0.0840567\end{array}\)

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

\({p_i} = \frac{{{d_i}}}{{200}}\)

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 鈥淪ample.鈥�
• Mark the values 0.00, 0.01, 0.02, 鈥︹��, 0.1 on the vertical axis and label the axis as 鈥淧roportion.鈥�
• Plot a straight line corresponding to the value 鈥�0.041667鈥� on the vertical axis and label the line (on the left side) as\(CL = 0.041667\).鈥�
• Plot a horizontal line corresponding to the value 鈥�0鈥� on the vertical axis and label the line as 鈥淟CL=0.鈥�
• Similarly, plot a horizontal line corresponding to the value 鈥�0.08405623鈥� on the vertical axis and label the line as 鈥淯CL=0.08405623.鈥�
• 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:

It can be observed from the plotted chart that there is at least one point that lies beyond the upper control limit.

Since the above criterion violates the stability of the given process, it can be concluded that the process is not within statistical control.

One value of the sample proportion is extremely high and lies beyond the upper control limit, and overall, there appears to be an increase in the proportion of defects. Therefore, the manufacturer should make appropriate changes, wherever required, to reduce the number of defects and ensure that good quality products are produced.