I am trying to understand a quiz question and I'm unsure of which distribution is involved and how to go about working this out.
The question reads.. Question
I know that the answer is 245 (as a result of guessing through the multiple choice option), but I don't know how to work this out. I assumed it is a poisson distribution question but I'm unsure.
The mean breaking strength cannot possibly be less than the specification minimum of 255 for the given failure rate.
I don't know why you assumed a Poisson distribution, but if we assume that it's normally distributed (which is usually a good default assumption in the absence of specific information), then I get a mean of 265.3 using Excel's NORMDIST in conjunction with GoalSeek.
In general, regardless of the actual distribution, the mean will be such that
$$ \int_{-\infty}^x p({\bf v},\xi)\text d\xi-r_f=0 $$
where ${\bf v}$ is the vector of distribution parameters that are related to the mean $\mu$ and variance $\sigma^2$, $x$ is the specification minimum (255 in the example) and $r_f$ is the reject rate (0.0197 in the example). Note that depending on the functional form of the cumulative distribution function, you may or may not need an iterative procedure to estimate $\mu$.