Question

Control Charts for p. In Exercises 5–12, use the given process data to construct a control chart for p. In each case, use the three out-of-control criteria listed near the beginning of this section and determine whether the process is within statistical control. If it is not, identify which of the three out-of-control criteria apply

Voting Rate In each of recent and consecutive years of presidential elections, 1000 people of voting age in the United States were randomly selected and the number who voted was determined, with the results listed below. Comment on the voting behavior of the population.

631 619 608 552 536 526 531 501 551 491 513 553 568

Short answer

The following p chart is constructed for the given proportion of voters:

The following features are observed from the chart:

• At least eight points lie below the central line.
• Points lie beyond UCL and LCL.

As a result, the process is not statistically controlled.

Furthermore, the actual voter turnout in the United States has increased in recent years, and actual statistics do not match those presented in the problem.

Step-by-step solution

Given information

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.

Important values of p chart

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:

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

Computation of the fraction of voters

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:

S.No.	Defectives (d)	Sample fraction of voters (p)
1	631	0.631
2	619	0.619
3	608	0.608
4	552	0.552
5	536	0.536
6	526	0.526
7	531	0.531
8	501	0.501
9	551	0.551
10	491	0.491
11	513	0.513
12	553	0.553
13	568	0.568

Construction of the p chart

Follow the given steps to construct the p chart:

• Mark the values 1, 2, ...,13on the horizontal axis and label it “Sample.”
• Mark the values 0.50, 0.52, 0.54, ……, 0.64 on the vertical axis and label it “Proportion.”
• Plot a straight line corresponding to the value 0.5523 on the vertical axis and label it (on the left side) “\(\bar P\;or\;\bar p = 0.5523\).”
• Plot a horizontal line corresponding to the value 0.5051 on the vertical axis and label it “LCL=0.5051.”
• Similarly, plot a horizontal line corresponding to the value 0.5995 on the vertical axis and label it “UCL=0.5995.”
• 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:

Analysis of the p chart

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.