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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:

$$ L = (8000 - A)\times 4 \quad \mbox{[ if } A< 8000\mbox{]}\quad\quad L = 0.5 \times (A-8000)\quad\mbox{[if } A> 8000\mbox{]} $$

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.

Probability distributions, expectation and variance. Spreadsheets. Expectation algebra. Confidence intervals. Mean. Experimental probability. Mathematical modelling. Frequency distribution. Inequality/inequalities. Normal distribution.