91影视

The number of defective ordersis given for 15 samples with a sample size of 50 orders each.

Here, the sample values show the number of defectives (which is an attribute) in each sample.

Thus, the appropriate control chart for depicting the proportion of defectives will be the p-chart which is an attribute chart.

It is computed as follows:

\(\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{observations}}}}\),

where\(\bar p\)isthe proportion of defective orders in all the samples and

\(\bar q = 1 - \bar p\).

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

\[\begin{aligned}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \end{aligned}\]

It is computed as follows:

\(\begin{aligned}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{observations}}}}\\ = \frac{{3 + 2 + 3 + ..... + 17}}{{15\left( {50} \right)}}\\ = 0.1613\end{aligned}\)

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

\(\begin{aligned}{c}\bar q = 1 - \bar p\\ = 1 - 0.161\\ = 0.839\end{aligned}\)

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

\[\begin{aligned}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.161 - 3\sqrt {\frac{{\left( {0.161} \right)\left( {0.839} \right)}}{{50}}} \\ = 0.0053\end{aligned}\]

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

\[\begin{aligned}{c}LCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.161 + 3\sqrt {\frac{{\left( {0.161} \right)\left( {0.839} \right)}}{{50}}} \\ = 0.3174\end{aligned}\]

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

\[{p_i} = \frac{{{d_i}}}{{50}}\],

where\[{p_i}\]isthe sample fraction defective for the ith lot, and

\[{d_i}\]isthe number of defective orders in the lot.

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

Follow the given steps to construct the p-chart:

• Mark the values 1, 2, ...,15 on the horizontal axis and label the axis as 鈥淒ay.鈥�
• Mark the values 0.00, 0.05, 0.10, 鈥︹��,0.35 on the vertical axis and label the axis as 鈥淧roportion.鈥�
• Plot a straight line corresponding to the value 鈥�0.1613鈥� on the vertical axis and label the line (on the left side) as 鈥淺(\bar P\;{\rm{or}}\;\bar p = 0.1613\).鈥�
• Plot a horizontal line corresponding to the value 鈥�0.0053鈥� on the vertical axis and label the line as 鈥淟CL=0.0053.鈥�
• Similarly, plot a horizontal line corresponding to the value 鈥�0.3174鈥� on the vertical axis and label the line as 鈥淯CL=0.3174.鈥�
• Mark the given15 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 features can be observed:

Here, the order times are increasing, so there is an upward trend in the proportion of defectives over the 15 days.

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

As these characteristics indicate the instability of the process, it can be concluded that the process is not within statistical control.