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The Wrong Stats

Age 16 to 18
Challenge Level

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

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.

Key questions

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?

Possible extension

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

Possible support

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?