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### Number and algebra

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### Advanced mathematics

# The Wrong Stats

### Why do this problem?

This
problem forces students to grapple with intuition in statistics
and to challenge modelling assumptions. This is important because
although statistical modelling is very powerful, knowledge of the
areas in which a model is likely to break down is critical to avoid
making significant predictive errors. This requires more than a
simple algebraic understanding of statistics.
### Possible approach

### Key questions

### Possible extension

### Possible support

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem is good for discussion in groups and not ideally
suited for individual use, since misconceptions are best uncovered
by describing them to other people.

For each part, can anyone suggest compelling reasons why the
assumptions cannot be true exactly? Can they suggest how flawed the
assumptions are (from 'completely wrong' to 'highly accurate in
practice')? Discuss the reasons in groups. The goal is that
EVERYONE agrees on the reasoning. It might be that some students
doubt the validity of an argument but don't feel confident enough
to voice their opinion. Encourage all doubts to be expressed, as
this will encourage the clearest thinking of all.

One point to be careful about is that students might try to
disclaim an answer via faulty statistical reasoning based on
another statistical pre-conception. For example, the first part on
the coin toss might be argued away by saying 'If we have had
several heads, then the chance of a tail can't remain the same'.
Listening carefully to the arguments will help to pick up such
errors. Note that this is a very positive aspect of this problem:
the more flawed statistical reasoning that the question challenges,
the better.

Does this part seem plausible to you?

Do you understand exactly the meaning of the technical
statistical language?

What 'extreme cases' of the situation might be considered to
test the validity of the assumption?

Creating similar statements is a really good way to come to
terms with the precise meaning of concepts in statistics. Could
students make statement involving the following concepts?

- Correlation
- Independence
- Poisson Distribution
- Binomial Distribution
- Continuous vs discrete random variables

In a group discussion there is always a useful role for the
careful listener. Perhaps those struggling could be used as a
critical audience for the explanations of others?