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:
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 $X$:
Don't forget that you can split an integral into two parts
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.