EX 4.18

The authors of “Assessment of Lifetime of Railway Axle” (Intl. J. of Fatigue, 2013: 40-46) used data collected from an experiment with a specified initial crack length and number of loading cycles to propose a normal distribution with

• mean value $5.496\mathrm{mm}$
• standard deviation $.067\mathrm{mm}$
• for the rv $X=$ final crack depth.

Question

For this model, what value of final crack depth would be exceeded by only $.5\%$ of all cracks under these circumstances?

Let $c$ denote the requested value. Then the desired condition is that $P\left(X>c\right)=.005$or, equivalently, that $P\left(X\le c\right)=.995.$ Thus $c$ is the 99.5th percentile of the normal distribution with $\mu =5.496$ and $\sigma =.067$. The 99.5th percentile of the standard normal distribution is 2.58, so

$$\begin{array}{rl}c& =\eta \left(0.995\right)\\ & =5.496+\left(2.58\right)\left(0.067\right)\\ & =5.496+0.173\\ & =5.669\text{mm}\end{array}$$

This is illustrated in Figure 4.23.

Figure 4.23 Distribution of final crack depth for Example 4.18