EXAMPLE 5.25

In a notched tensile fatigue test on a titanium specimen, the expected number of cycles to first acoustic emission (used to indicate crack initiation) is $\mu =28,000$ , and the standard deviation of the number of cycles is $\sigma =5000$ . Let $X_1,X_2,\dots ,X_{25}$ be a random sample of size 25, where each $X_i$ is the number of cycles on a different randomly selected specimen. Then the expected value of the sample mean number of cycles until first emission is $E\left(\bar{X}\right)=\mu =28,000$ , and the expected total number of cycles for the 25 specimens is $E\left(T_0\right)=n\mu =25\left(28,000\right)=700,000$ . The standard deviation of $\bar{X}$ (standard error of the mean) and of $T_0$ are

$${\sigma }_{\bar{X}}=\sigma /\sqrt{n}=\frac{5000}{\sqrt{25}}=1000$$ $${\sigma }_{T_o}=\sqrt{n}\sigma =\sqrt{25}\left(5000\right)=25,000$$

If the sample size increases to $n=100,E\left(\bar{X}\right)$ is unchanged, but ${\sigma }_{\bar{X}}=500$ , half of its previous value (the sample size must be quadrupled to halve the standard deviation of $\bar{X}$ ).