Challenging +1.2 This question requires understanding of linear combinations of normal random variables and scaling properties, involving multiple steps: forming expressions for costs (linear functions of normals), finding distributions for multi-day totals, computing the distribution of their difference, and calculating a probability. While conceptually straightforward for Further Maths Statistics students, it requires careful algebraic manipulation and correct application of variance rules for independent normals, placing it moderately above average difficulty.
5.
A company uses two drivers for deliveries.
Driver \(A\) charges a fixed rate of \(\pounds 80\) per day plus \(\pounds 2\) per mile travelled on that day.
Driver \(B\) charges a fixed rate of \(\pounds 120\) per day plus \(\pounds 1.50\) per mile travelled on that day.
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution \(\mathrm { N } ( 83,360 )\).
Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver \(A\) is at least \(\pounds 300\) more than the total charge of two randomly chosen days' deliveries by driver \(B\).
5.
A company uses two drivers for deliveries.\\
Driver $A$ charges a fixed rate of $\pounds 80$ per day plus $\pounds 2$ per mile travelled on that day.\\
Driver $B$ charges a fixed rate of $\pounds 120$ per day plus $\pounds 1.50$ per mile travelled on that day.\\
On each working day the total distance, in miles, travelled by each driver is a random variable with the distribution $\mathrm { N } ( 83,360 )$.
Find the probability that the total charge to the company of three randomly chosen days' deliveries by driver $A$ is at least $\pounds 300$ more than the total charge of two randomly chosen days' deliveries by driver $B$.\\
\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q5 [6]}}