Question

Flaws in the plastic-coated wire. The British Columbia Institute of Technology provides on its Web site (www.math.bcit.ca) practical applications of statistics at mechanical engineering firms. The following is a Poisson application. A roll of plastic-coated wire has an average of .8 flaws per 4-meter length of wire. Suppose a quality-control engineer will sample a 4-meter length of wire from a roll of wire 220 meters in length. If no flaws are found in the sample, the engineer will accept the entire roll of wire. What is the probability that the roll will be rejected? What assumption did you make to find this probability?

Short answer

The probability that the roll will be rejected is 0.5507.

Step-by-step solution

Given information

There is a roll of plastic-coated wire that has an average of 0.8 flaws per 4-meter length of wire.

x be the number of flaws in a 4-meter length of wire

Finding the probability

$x ~ Poisson\left(\lambda \right)$where$\lambda =0.8$

The probability mass function of x is

$P\left(x\right)=\frac{e^{-\lambda }{\lambda }^x}{x!}$

Roll will be rejected if there is at least one flaw in the sample of a 4-meter length of wire.

The probability is,

$\begin{array}{rcl}P\left(x⩾1\right)& =& 1-P\left(x=0\right)\\ & =& 1-\frac{e^{-0.8}{0.8}^0}{0!}\\ & =& 1-0.4493\\ & =& 0.5507\\ & & \end{array}$

$P\left(x⩾1\right)=0.5507$

Therefore, the probability that the roll will be rejected is 0.5507.

Since the process has a Poisson distribution, we have to assume that the flaws are randomly distributed, and the 4-meter length of sample wire represents the entire roll.