Question

The no. of defects in 20 pieces of cloth each of 100 meter length
is given below:
2, 1, 3, 3, 1, 6, 4, 3, 7, 10, 2, 2, 6, 4, 3, 2, 1, 5, 6, 4.
Draw the appropriate control chart and interpret the result.

Answer 1

Answer on Question #55468 – Math – Statistics and Probability

The no. of defects in 20 pieces of cloth each of 100 meter length is given below:

2, 1, 3, 3, 1, 6, 4, 3, 7, 10, 2, 2, 6, 4, 3, 2, 1, 5, 6, 4.

Draw the appropriate control chart and interpret the result.

Solution

A Control chart is made for decision making by management reviewing the value of statistic if it is outside the threshold limits.

The statistic here is number of defects: $x$. We have $N = 20$.

Average of $x$ is $\bar{x} = \frac{\sum x}{N} = \frac{75}{20} = 3.75$

Standard deviation of $x$ is $s = 2.657$

Standard error is $SE = \frac{s}{\sqrt{20}} = 0.5916$

Warning levels for $x$:

$$x = \bar{x} + 2SE = 4.9332 \quad \text{and} \quad x = \bar{x} - 2SE = 2.5668$$

A piece is accepted with a warning if

$$\bar{x} - 3SE < x < \bar{x} - 2SE \quad \text{or} \quad \bar{x} + 2SE < x < \bar{x} + 3SE$$

Accepted range:

$$\bar{x} - 3SE < x < \bar{x} + 3SE$$ $$1.9752 < x < 5.5248$$

Rejection threshold levels: control limits.

rejected if: $x > \bar{x} + 3SE$ or $x < \bar{x} - 3SE$

So the pieces with number of defects: 1, 6, 7, 10, will be sent for quality assurance and process verification, the cause analysis.

Normally, for a good process, $99.7\%$ of the values should be between the control limits. Here we have 8 values of 20 being outside the control range.

The manufacturing process needs to be reviewed.

www.AssignmentExpert.com