91影视

Data are given on the number of voters in 13 randomly selected samples of the people of voting age.

The size of each sample is 1000.

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

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{observations}}}}\\ = \frac{{631 + 618 + 608 + ..... + 568}}{{13\left( {1000} \right)}}\\ = \frac{{7180}}{{13000}}\\ = 0.5523\end{array}\)

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

\(\begin{array}{c}\bar q = 1 - 0.5523\\ = 0.4477\end{array}\)

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

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.5523 - 3\sqrt {\frac{{\left( {0.5523} \right)\left( {0.4477} \right)}}{{1000}}} \\ = 0.5051\end{array}\)

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

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.5523 + 3\sqrt {\frac{{\left( {0.5523} \right)\left( {0.4477} \right)}}{{1000}}} \\ = 0.5995\end{array}\)

The sample fraction of voters for the ith region can be computed as shown below:

\({p_i} = \frac{{{d_i}}}{{1000}}\)

Here,

\({p_i}\)isthe sample fraction of voters for the ith region, and

\({d_i}\)isthe number of voters in the ith region.

The computation of fraction of voters for the ith region is given as follows:

Follow the given steps to construct the p chart:

• Mark the values 1, 2, ...,13on the horizontal axis and label it 鈥淪ample.鈥�
• Mark the values 0.50, 0.52, 0.54, 鈥︹��, 0.64 on the vertical axis and label it 鈥淧roportion.鈥�
• Plot a straight line corresponding to the value 0.5523 on the vertical axis and label it (on the left side) 鈥淺(\bar P\;or\;\bar p = 0.5523\).鈥�
• Plot a horizontal line corresponding to the value 0.5051 on the vertical axis and label it 鈥淟CL=0.5051.鈥�
• Similarly, plot a horizontal line corresponding to the value 0.5995 on the vertical axis and label it 鈥淯CL=0.5995.鈥�
• Mark all13 sample points (number of defectives of the ith region) on the graph and join the dots using straight lines.

The following p chart is obtained:

The following featuresare observed from the chart:

• There are at least eight points that lie below the central line.
• There is at least one point that lies beyond the upper control limit (UCL).
• There is at least one point that lies below the lower control limit (LCL).

Thus, the criteria indicate that the process is not within statistical control.

Moreover, the actual voter turnout in the US has increased in recent years.The actual values are not close to those given in the problem.