91影视

Data are given on the proportion of defective coins in 10 samples.

The sample size of each sample is 10,000.

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

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{samples}}}}\\ = \frac{{32 + 21 + 25 + ..... + 33}}{{10\left( {10000} \right)}}\\ = \frac{{282}}{{100000}}\\ = 0.00282\end{array}\)

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

\(\begin{array}{c}\bar q = 1 - 0.00282\\ = 0.99718\end{array}\)

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

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.00282 - 3\sqrt {\frac{{\left( {0.00282} \right)\left( {0.99718} \right)}}{{10000}}} \\ = 0.001229\end{array}\)

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

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.00282 + 3\sqrt {\frac{{\left( {0.00282} \right)\left( {0.99718} \right)}}{{10000}}} \\ = 0.004411\end{array}\)

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

\({p_i} = \frac{{{d_i}}}{{10000}}\)

Here,

\({p_i}\)isthe 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 1, 2, ...,10 on the horizontal axis and label it 鈥淪ample.鈥�
• Mark the values 0.0010, 0.0015, 0.0020, 鈥︹��, 0.0045 on the vertical axis and label it鈥淧roportion.鈥�
• Plot a straight line corresponding to the value 0.0093 on the vertical axis and label it (on the left side) 鈥淺(\bar P\;or\;\bar p = 0.00282\).鈥�
• Plot a horizontal line corresponding to the value 0.001229 on the vertical axis and label it鈥淟CL=0.001229.鈥�
• Similarly, plot a horizontal line corresponding to the value 0.004411 on the vertical axis and label it UCL=0.004411.鈥�
• Mark all ten sample points (fraction defective of the ith lot) on the graph and join the dots using straight lines.

The following p chart is obtained:

None of the three out-of-control criteria is depicted in the plotted p chart.

As all the points lie within upper and lower control limits, it can be concluded that the process is within statistical control.