A quality analyst wants to construct a control chart for determining whether four machines, all producing the same product, are in 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: $$\begin{array} { c c }
\text { Machine } & \text { Total Defectives } \
\hline \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$
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 in 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: $$\begin{array} { c c } 
\text { Machine } & \text { Total Defectives } \
\hline \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$
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 need to calculate the sample proportion of defectives for each machine and plot them on the chart. The control limits are calculated using the following formula:
Upper Control Limit: p̂ + 3√(p̂(1-p̂)/n)
Lower Control Limit: p̂ - 3√(p̂(1-p̂)/n)
where p̂ is the overall sample proportion of defectives (computed by adding up the defectives from all machines and dividing by the total sample size) and n is the sample size for each machine.

Here are the relevant calculations:

- Total defectives: 23 + 15 + 29 + 13 = 80
- Total sample size: 4,000 (1,000 for each machine)
- Overall sample proportion of defectives: 80/4,000 = 0.02
- Control limits:
Upper = 0.02 + 3√(0.02(1-0.02)/1,000) = 0.026
Lower = 0.02 - 3√(0.02(1-0.02)/1,000) = 0.014

Now let's calculate the sample proportion of defectives for each machine:

- Machine 1: 23/1,000 = 0.023 -> within control limits
- Machine 2: 15/1,000 = 0.015 -> within control limits
- Machine 3: 29/1,000 = 0.029 -> above upper control limit
- Machine 4: 13/1,000 = 0.013 -> within control limits

Therefore, machines #3 and #4 appear to be out-of-control for process proportion of defectives, but machine #4 is only slightly below the upper control limit and may be worth monitoring further before taking any action.

A Quality Analyst Wants to Construct a Control Chart for Determining