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: $$\begin{array} { c c }
\text { Machine } & \text { Total Defectives } \
\hline \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$
What are the control chart upper and lower control limits for an alpha risk of .05?
A) .0272 and .0128
B) .0287 and .0113
C) .029 and .013
D) .0303 and .0097
E) .0332 and .0068

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: $$\begin{array} { c c } 
\text { Machine } & \text { Total Defectives } \
\hline \# 1 & 23 \
\# 2 & 15 \
\# 3 & 29 \
\# 4 & 13
\end{array}$$
What are the control chart upper and lower control limits for an alpha risk of .05?
A) .0272 and .0128
B) .0287 and .0113
C) .029 and .013
D) .0303 and .0097
E) .0332 and .0068

Quizplus · Accepted Answer

To construct the control chart, we first need to calculate the overall proportion of defectives in the sample:

$$p=\frac{	ext{Total Defectives}}{	ext{Total Sample Size}}=\frac{23+15+29+13}{4,000}=0.020$$

Next, we need to calculate the standard deviation of the sampling distribution of the proportion of defectives. Since the sample sizes are all 1,000, we can use the formula:

$$\sigma=\sqrt{\frac{pq}{n}}=\sqrt{\frac{(0.020)(1-0.020)}{1,000}}=0.00436$$

where $q=1-p$.

Finally, we can calculate the control chart limits using the following formulas:

$$	ext{UCL}=p+3\sigma=0.0287$$

$$	ext{LCL}=p-3\sigma=0.0113$$

Therefore, the answer is B.

A Quality Analyst Wants to Construct a Control Chart for Determining