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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Aim High

### 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

### Key questions

### Possible extension

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Teachers' Resources

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.

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?

Parts 2 and 3 of this question naturally offer
extension.

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.