The duration, $X$ hours, that students spend using a games machine on any one day follows a normal distribution with mean $3.24$ hours and standard deviation $0.96$ hours.
Over the $365$ days of the year, how many days would you expect a randomly selected student to spend less than $4$ hours using a games machine?
Determine the value of $k$ for which $\mathrm{P}(X > k) = 0.2$.
Find the probability that a randomly chosen student uses a games machine for a number of hours in a day that lies within $1.5$ standard deviations of the mean.
The random variable $Y$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$, where $4\sigma = 3\mu$ and $\mu \neq 0$. Find the probability that a randomly selected value of $Y$ is greater than $0$.