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Why do this problem?

This problem offers the opportunity for a statistical investigation of confidence intervals and expectations at various levels of sophistication.

Possible approach

There are three possible levels at which this problem can be tackled, and how it is used depends on the levels of the students. As an exercise in confidence intervals it is relatively simple from a calculational point of view; the interest lies in making the right decisions for the overall final planting recommendation. There are various ways in which students might do this, from making a few sensible guesses and extrapolating, to doing a numerical investigation.

The second parts in which the expectations are found are more difficult. They will require a sophistication in manipulating integrals, changing variables in integrals and using the Erf or CumNorm functions. This would be a good task to use in this way to draw together many strands from a study of statistics.

Key questions

Why can a yield never be modelled exactly by a normal distribution?
How are the sums of 2, 3, 4, ... identical normal distributions distributed?
Why is the mean target of 8000 tonnes clearly not a good one?
What is the formula for expectation in terms of an integral?

Possible extension

Parts 2 and 3 of this question naturally offer extension.

Possible support

Focus only on the first part of the question, which forms a nice self-contained task. However, it is worth discussing with students that methods exist to take the analysis to a higher level.