I've been trying to understand how to go around the third requirement in this problem:
This is how its solved on Slader:
How do we get info about whether the $30\%$ figure applies by assuming that that's the case and then finding the probability of getting exactly five failures for our new observation (plant), it's already clear that in the new observation $25\%$ of the failures are due to operator error. I can't seem to draw the link between doing this helps know whether the $30\%$ figure applies or not.
Thank you.
The explanation given in the image is a little hand-wavey, but the overall idea is sound. To explain why, suppose instead that $20$ out of $20$ of the failures were caused by operator errors. If the theoretical probability of an operator error really is $30 \%$, then the chance of this occuring is $$ 0.3^{20}=\text{very small number...} $$ From a practical perspective, what seems more likely: that (a) this was simply a freak event, and that the probability of an operator error really is $30\%$; or (b) the $30\%$ figure is not accurate. I think you will undoubtedly agree that (b) is the more sensible option.
Now let's consider your example: if the theoretical probability of an operator error is $30\%$, then getting $5$ out of $20$ failures stemming from operator errors does not seem out of the ordinary. We have no reason to suspect that the $30\%$ is inaccurate.
To do this more rigorously, you can conduct a hypothesis test. We'll consider a batch of $20$ errors, and agree beforehand that if, say, $10$ or more failures are caused by operator errors, we will conclude that there is sufficient evidence to suggest that the $30\%$ figure is inaccurate. This seems reasonable: after all, if the theoretical probability of an operator error is $30\%$, then the chance of getting $10$ or more failures being caused by operator errors is only about $4.7\%$. Again, it's a question of which interpretation of the events is more reasonable: that the $30\%$ is accurate, and we just experienced an unlikely event; or that the events witnessed are enough evidence to suggest that the $30\%$ figure is not right.
Note that context is really important here. If we were very confident that the chance of an operator error being the cause of failure really is $30\%$, then we will need a lot of evidence to overturn our beliefs. This might mean, for example, that we will only conclude that the $30\%$ figure is innaccurate if $12$ or more failures are caused by an operator error.