91影视

Data are given on the number of defects in 20 randomly selected samples.

The size of each sample is 50.

Let\(\bar p\)be the estimated proportion of defective cans in all samples.

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{observations}}}}\\ = \frac{{8 + 7 + 9 + ..... + 7}}{{20\left( {50} \right)}}\\ = \frac{{165}}{{1000}}\\ = 0.165\end{array}\)

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

\(\begin{array}{c}\bar q = 1 - 0.165\\ = 0.835\end{array}\)

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

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.165 - 3\sqrt {\frac{{\left( {0.165} \right)\left( {0.835} \right)}}{{50}}} \\ = 0.0075\end{array}\)

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

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.165 + 3\sqrt {\frac{{\left( {0.165} \right)\left( {0.835} \right)}}{{50}}} \\ = 0.3225\end{array}\)

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

\({p_i} = \frac{{{d_i}}}{{50}}\)

Here,

\({p_i}\)is the sample fraction defective for the ith batch, and

\({d_i}\)is 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 2, 4, ...,20on the horizontal axis and label it 鈥淪ample.鈥�
• Mark the values 0.00, 0.05, 0.10, 鈥︹��, 0.35 on the vertical axis and label it 鈥淧roportion.鈥�
• Plot a straight line corresponding to the value 0.165 on the vertical axis and label it (on the left side) 鈥淺(\bar P\;or\;\bar p = 0.165\).鈥�
• Plot a horizontal line corresponding to the value 0.0075 on the vertical axis and label it 鈥淟CL=0.0075.鈥�
• Similarly, plot a horizontal line corresponding to the value 0.3225 on the vertical axis and label it 鈥淯CL=0.3225.鈥�
• Mark all 20 sample points (fraction defective of the ith lot) on the graph and join the dots using straight lines.

The following p chart is obtained:

The chart has no feature that indicates the violation of the stability of the process. Thus, the process is within statistical control.

But the values of the proportions of defectives seem to be very high.

Thus, the manufacturer should take quick action to correct the quality of the goods produced.