A random variable $X$ represents the number of hours of use until an electronic component needs to be replaced. $X$ is normally distributed with a mean of $27500$ hours and a standard deviation of $4000$ hours. $99$% of the components do not need to be replaced until they have been used for $k$ hours. Find the value of $k$.
The answer in the mark scheme is $36000$. How come? Shouldn't $k$ be below the mean as there is $99$% of devices that still work? I calculated it from the other side of the mean, as I think I need a small number of hours so that $99$% still work. My answer was approximately $18000$.
No, the $k$ you are after is referring to the $99$th percentile, as depicted below, which is clearly above the mean $0$, and the shaded area refers to the $36000$ hours your classmates got.
A better reading is thus that
the last of $99\%$ of the components does not need to be replaced until it has been used for $k$ hours. So after $k$ hours, only $1\%$ of all initial devices are still working.