Aim high
Problem
A farmer has a very large farm which produces wheat. The yield of wheat per hectare is known to be normally distributed at $7.74$ tonnes per hectare, with a standard deviation of $0.62$.
Why can a normal distribution of a yield never be entirely accurate? Why, in this case, does it not matter?
It is nearly planting time and the farmer has future orders for $8000$ tonnes of wheat. If he fails to produce enough wheat then he will have to pay a stiff penalty to the buyer; if he produces too much wheat then he will have to offload or destroy the surplus at a loss. In each case, the loss $L$ can be modelled as follows, where $A$ is the amount he plants:
$$\begin{align*} &\mbox{If } A < 8000, L = (8000 - A)\times 4\\
&\mbox{If } A > 8000, L = 0.5 \times (A-8000)\\\end{align*}$$
There are three possible levels of analysis of this problem
Level 1 - using confidence intervals
How much wheat would the farmer have to plant to have an expected yield of exactly $8000$ tonnes? What distribution would the total yield have, and what would be the $95\%$ confidence interval for the actual yield?
In this case, what would be the $95\%$ confidence interval for the loss?
How would this alter if he instead planted $1000$ hectares? Could you recommend an ideal amount of wheat to plant?
Level 2 - finding an expected loss
For your planting recommendation, what would be the expected loss?
Level 3 - minimising the expected loss (involves difficult, but fun, calculus)
How much should the farmer plant to minimise his expected loss? Make an initial considered estimate before performing a calculation.
Getting Started
For the second part you will need to use the formula for the expected value of a random variable $X$:
$$E(X) = \int^\infty_{-\infty} xf(x) dx$$
Don't forget that you can split an integral into two parts
$$\int^\infty_{-\infty} xf(x) dx=\int^a_{-\infty} xf(x) dx+\int^\infty_a xf(x) dx$$
You might also want to use the Erf function in Excel to work out the numbers and to change variables in any integrals to reduce them to standard $N(0,1)$ cases.