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Why do this problem?
will get students into statistics without them necessarily
needing to engage in detailed calculation (although some students,
depending on their level, might wish to calculate various
statistical quantities). It will lead to a better intuitive grasp
of statistics, which will inevitably lead to a better grasp of
various statistical techniques. The questions are approximately
listed in order of difficulty.
This problem is very well suited to discussion. As there are
no numbers, it is very easy for all students to try to get into
this problem at a level which suits them. One approach might be to
ask for an immediate, instinctive response to the questions before
asking them to assess them in more detail. Were their gut-feelings
right or wrong? Are there any surprises?
This problem offers a good chance to practise explaining
complicated ideas in statistics. Students could try to explain
their thoughts verbally to each other. To give a good explanation
would require a sound analysis of the statistics. Does the audience
think that the explanation is sound or convincing?
To give a sound analysis will require some quantification of the
concepts of the key words: 'sometimes', 'always', or 'never'.
This question gives an opportunity to explore the power of
counter-examples in a mathematical analysis: for example,
constructing a single example in which 'Half of the students taking
a test DONT score less than the average mark' shows that the
statement 'Half of the students taking a test score less than the
average mark' cannot ALWAYS be true.
Assessing the meanings of 'sometimes' and 'nearly always' will be
more open to discussion. This could easily lead to discussions of
normal distributions, statistical testing and confidence
It would be good to use this problem at an early (intuitive) stage
in the study of statistics and then revisit it towards the end of a
course of statistics (once computation skills are developed).
Comparison of answers at these two stages would be an interesting
It is important to note that this problem is likely to raise many
questions (such as the meaning of the word 'average'). All
questions are valid and exploration of the issues raised
will lead to a stronger intuitive understanding of statistics,
which can only impact positively the subsequent learning of more
formal statistical techniques.
This problem may feel very 'open' to certain students.
Questions should be chosen to encourage students to think their way
into each statement and to consider possible, concrete scenarios in
which each one might or might not be true.
The last two parts of this question are challenging in
themselves. For further work on these statistical ideas, you might
refer students to Random
, in which bounds on statistical quantities are
Students could focus on the first four parts of the question
and invent scores for the people in their class and then see how
the averages work out. You might also refer to the problem Misunderstanding