A quality analyst wants to construct a control chart for determining whether four machines, all producing the same product, are under control with regard to a particular quality attribute. Accordingly, she inspected 1,000 units of output from each machine in random samples, with the following results: $$\begin{array} { l c }
\text { Machine } & \text { Total Defectives } \
\hline \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$ For upper and lower control limits of .026 and .014, which machine(s), if any, appear(s) to be out-of-control for process proportion of defectives?
A) machine #3
B) machine #4
C) machines #3 and #4
D) machines #2 and #3
E) none of the machines

Question

A quality analyst wants to construct a control chart for determining whether four machines, all producing the same product, are under control with regard to a particular quality attribute. Accordingly, she inspected 1,000 units of output from each machine in random samples, with the following results: $$\begin{array} { l c } 
\text { Machine } & \text { Total Defectives } \
\hline   \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$ For upper and lower control limits of .026 and .014, which machine(s), if any, appear(s) to be out-of-control for process proportion of defectives?
A) machine #3
B) machine #4
C) machines #3 and #4
D) machines #2 and #3
E) none of the machines

Quizplus · Accepted Answer

To construct the control chart, we first calculate the overall proportion of defectives in all machines combined:
$$p=\frac{	ext{Total defectives}}{	ext{Total units inspected}}=\frac{23+15+29+13}{4,000}=0.020$$
Then, we can calculate the standard deviation of the proportion of defectives as:
$$\sigma=\sqrt{\frac{pq}{n}}=\sqrt{\frac{(0.020)(1-0.020)}{1,000}}\approx0.00436$$
where $p$ is the overall proportion of defectives, $q=1-p$, and $n$ is the sample size (1,000 units inspected for each machine).

Using the upper and lower control limits given:
$$	ext{UCL}=p+z\sigma=0.026$$
$$	ext{LCL}=p-z\sigma=0.014$$
where $z$ is the appropriate z-score for the desired confidence level and sample size. For example, for a 99.7% confidence level (corresponding to a 3-sigma control chart), $z=3$.

Now, we can calculate the proportion of defectives for each machine and determine whether any of them exceed the control limits:
$$\begin{array}{c|c|c|c}
	ext{Machine} & 	ext{Defectives} & 	ext{Sample size} & 	ext{Proportion} \
\hline
1 & 23 & 1,000 & 0.023 \
2 & 15 & 1,000 & 0.015 \
3 & 29 & 1,000 & 0.029 \
4 & 13 & 1,000 & 0.013
\end{array}$$
Only machine #3 has a proportion of defectives above the UCL, and both machines #3 and #4 have proportions below the LCL. Therefore, we can conclude that machines #3 and #4 are out-of-control for process proportion of defectives. The correct answer is C.

A Quality Analyst Wants to Construct a Control Chart for Determining