Question

A bag contains rubber bands with lengths that are normally distributed with a mean of 6 cm of length, and a standard deviation of 1.5 cm. What percentage of rubber bands can be expected to be shorter than 3 cm?

Answer 1

Let's say \(X\) denotes the random variable for rubber band length. Transform \(X\) to \(Z\), which follows the standard normal distribution, via \(\dfrac{X-\mu}{\sigma}\), where \(\mu\) and \(\sigma\) are the mean and standard deviation for \(X\).
\[\mathbb P(X<3)=\mathbb P\left(\frac{X-6}{1.5}<\frac{3-6}{1.5}\right)=\mathbb P(Z<-2)\]If you know the empirical (68–95–99.7) rule, then you know that approximately \(95\%\) of the distribution falls within two standard deviations of the mean, i.e. \(\mathbb P(|Z|<2)\approx0.95\).

Then this means \(\mathbb P(|Z|>2)=\mathbb P((Z>2)\cup(Z<-2))\approx1-0.95=0.05\).

Finally, because the distribution is symmetric, you have \(\mathbb P(Z>2)=\mathbb P(Z<-2)\). These events are mutually exclusive, so
\[\begin{align*}
\mathbb P((Z>2)\cup(Z<-2))&=\mathbb P(Z>2)+\mathbb P(Z<-2)\\[1ex]
0.05&\approx2\mathbb P(Z<-2)\\[1ex]
\mathbb P(Z<-2)&\approx0.025
\end{align*}\]This assumes you don't need a super-precise answer, but it's really not that far off: http://www.wolframalpha.com/input/?i=p(z%3C-2)