Question

Notation The control chart for Exercise 1 shows a value of $\overline{p}$ = 0.0975. What does that value denote, and how is it obtained? What do UCL and LCL indicate?

Short answer

The value of$\overline{p}$denotes the estimated proportion of defectives in all the sample observations.

The value of$\overline{p}$is obtained using the given formula:

$\overline{p}=\frac{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}}$

UCL stands for the upper control limit, and LCL stands for lower control limit. They indicate the limits which form the boundary within which the proportion of defective quarters should fall.

Step-by-step solution

Given information

A p-chart is plotted to examine if the process is under control or not.

Here, the p-chart depicts the proportion of defective quarters manufactured for a series of times.

Value of $\overline{p}$

On referring to the p-chart, the value of$\overline{p}$is obtained to be equal to 0.0975.

The value of$\overline{p}$indicates the estimated proportion of defectives in the entire process.

It is obtained using the given formula:

$\overline{p}=\frac{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}}$

Step 3:Identify the UCL and LCL

Here, UCL stands for upper control limit and LCL stands for lower control limit.

The limits define a boundary within which the value of the characteristic under study must fall so that the process is statistically stable.

Here, the value of UCL is equal to 0.1865, and the value of LCL is equal to 0.0085. This implies that the value of the proportion of defects should fall within these limits for the process to be under statistical control.