A quality analyst wants to construct a control chart for determining whether three machines, all producing the same product, are in control with regard to a particular quality variable. Accordingly, he sampled four units of output from each machine, with the following results: $$\begin{array} { c c c c c }
\text { Machine } && { \text { Measurements } } \
\hline \# 1 & 17 & 15 & 15 & 17 \
\# 2 & 16 & 25 & 18 & 25 \
\# 3 & 23 & 24 & 23 & 22
\end{array}$$
What are the x-bar chart three-sigma upper and lower control limits?
A) 22 and 18
B) 23.29 and 16.71
C) 23.5 and 16.5
D) 23.16 and 16.84
E) 24 and 16

Question

A quality analyst wants to construct a control chart for determining whether three machines, all producing the same product, are in control with regard to a particular quality variable. Accordingly, he sampled four units of output from each machine, with the following results: $$\begin{array} { c c c c c } 
\text { Machine } && { \text { Measurements } } \
\hline \# 1 & 17 & 15 & 15 & 17 \
\# 2 & 16 & 25 & 18 & 25 \
\# 3 & 23 & 24 & 23 & 22
\end{array}$$
What are the x-bar chart three-sigma upper and lower control limits?
A) 22 and 18
B) 23.29 and 16.71
C) 23.5 and 16.5
D) 23.16 and 16.84
E) 24 and 16

Quizplus · Accepted Answer

1. Calculate the average of each machine:
$\bar{x}_1 = \frac{17+15+15+17}{4} = 16$
$\bar{x}_2 = \frac{16+25+18+25}{4} = 21$
$\bar{x}_3 = \frac{23+24+23+22}{4} = 23$
2. Calculate the overall mean:
$\bar{\bar{x}}=\frac{\sum\bar{x}}{3}=\frac{16+21+23}{3}=20$
3. Calculate the overall range:
$R=	ext{max}(	ext{data})-	ext{min}(	ext{data})=25-15=10$
4. Calculate the control limits:
$UCL=\bar{\bar{x}}+ \frac{3}{d_2} R=\bar{\bar{x}}+\frac{3}{0.729} \frac{R}{4}=20+\frac{3}{0.729} \frac{10}{4}=23.16$
$LCL=\bar{\bar{x}}- \frac{3}{d_2} R=\bar{\bar{x}}-\frac{3}{0.729} \frac{R}{4}=20-\frac{3}{0.729} \frac{10}{4}=16.84$
Therefore, the x-bar chart three-sigma upper and lower control limits are 23.16 and 16.84, respectively. The answer is D.

A Quality Analyst Wants to Construct a Control Chart for Determining