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

Euro Coins Repeat Exercise 5, assuming that the size of each batch is 100 instead of 10,000. Compare the control chart to the one found for Exercise 5. Comment on the general quality of the manufacturing process described in Exercise 5 compared to the manufacturing process described in this exercise.

Short answer

The following p chart is constructed for the proportions of defective coins:

The process seems to be within statistical control since none of the three out-of-control criteria is visible from the chart.

The p chart corresponding to the sample size of 10,000 is nearly identical to the p chart shown above with a sample size of 100.

Because the two p charts have comparable outlooks, the manufacturing process quality for the two processes can also be considered to be the same.

Furthermore, whether the sample size is 100 or 10,000, the process produces good and reliable goods, and the manufacturer should continue with the current procedure.

Step-by-step solution

Given information

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

The sample size of each of the 10 samples is equal to 100.

Important values of p chart

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

It is computed as follows:

\(\begin{array}{c}\bar p = \frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{samples}}}}\\ = \frac{{32 + 21 + 25 + ..... + 33}}{{10\left( {100} \right)}}\\ = \frac{{282}}{{1000}}\\ = 0.282\end{array}\)

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

\(\begin{array}{c}\bar q = 1 - 0.282\\ = 0.718\end{array}\)

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

\(\begin{array}{c}LCL = \bar p - 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.282 - 3\sqrt {\frac{{\left( {0.282} \right)\left( {0.718} \right)}}{{100}}} \\ = 0.147008\end{array}\)

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

\(\begin{array}{c}UCL = \bar p + 3\sqrt {\frac{{\bar p\bar q}}{n}} \\ = 0.282 + 3\sqrt {\frac{{\left( {0.282} \right)\left( {0.718} \right)}}{{100}}} \\ = 0.416992\end{array}\)

Computation of the fraction defective

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

\({p_i} = \frac{{{d_i}}}{{100}}\)

Where,

\({p_i}\)be the sample fraction defective for the ith batch;

\({d_i}\)be the number of defective orders in the ith batch.

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

S.No.	Defectives (d)	Sample fraction defective (p)
1	32	0.32
2	21	0.21
3	25	0.25
4	19	0.19
5	35	0.35
6	34	0.34
7	27	0.27
8	30	0.30
9	26	0.26
10	33	0.33

Construction of the p chart

Follow the given steps to construct the p chart:

• Mark the values 1, 2, ..., 10 on the horizontal axis and label the axis as “Sample.”
• Mark the values 0, 0.05, 0.1, ……, 0.45 on the vertical axis and label the axis as “Proportion.”
• Plot a straight line corresponding to the value “0.282” on the vertical axis and label the line (on the left side) as “\(CL = 0.282\).”
• Plot a horizontal line corresponding to the value “0. 147008” on the vertical axis and label the line as “LCL=0.147008”.
• Similarly, plot a horizontal line corresponding to the value “0.416992” on the vertical axis and label the line as “UCL=0.416992.”
• Mark the given 10 sample points (fraction defective of the ith batch) on the graph and join the dots using straight lines.

The following p chart is plotted:

Analysis of the p chart

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

Thus, it can be concluded that the process is within statistical control.

Comparison

Referring to Exercise 5, the p chart corresponding to each sample size is equal to 10,000. In this exercise, the p chart with a sample size equal to 100 is approximately identical to the referred Exercise 5.

Since the two p charts have similar outlooks, it can be said that the quality of the manufacturing process for the two processes is also the same.

Moreover, the process dishes out good and reliable products (whether the sample size is 100 or 10,000), and the manufacturer should continue with the ongoing procedure.