
Required for next week

Reading
     ``Statistical evidence of cheating on multiple choice tests.'' S.P.
     Klein (1992), Chance 5 (3-4), 23--27. and `` The use of statistical
     evidence in allegations of exam cheating.'' P.J. Boland and M. Proschan
     (1990), Chance 3 (3), 10--14.
Your task
     Come prepared to discuss these articles, and to ask questions about the
     parts you didn't understand.

Technical note: testing hypotheses

An extremely common form of statistical inference is an ``hypothesis test'',
or a ``significance test''. (There is in fact a technical difference between
hypothesis tests and significance tests, but we needn't worry about that
here.) It is used to answer the question ``Are these data consistent with
-------- ?'', where -------- is a formulation of an hypothesis. Usually, the
formulation will be a mathematical one, relating to some chance mechansism,
i.e. probability model, that may have generated the data.

In the context of the streak shooting article TG, the hypothesis that the
authors considered was that the sequence of hits and misses for a given
player was just like a sequence of independent coin tosses, with the
probability of a hit equal to the player's overall hit rate. In other words,
their hypothesis was ``there is no streak shooting''.

Their hypothesis testing procedure is outlined in the last paragraph on p.18
and the first two columns on p.19. For each player in Table 2, they took
strings of four shots, and counted for each string whether it had 3 or 4
hits (high), 2 hits (moderate), or 0 or 1 hit (low). If successive shots are
like coin tosses, then the probabilities for high, moderate, and low can be
easily calculated for each player. For Clint Richardson, who has an overall
hit rate of 0.50, these probabilities are 5/16, 6/16, 5/16, respectively.
They don't give us the data to reproduce their calculations, but what they
did next was to compare the observed counts of high, medium and low to these
predictions. Their conclusion was, yes, the data is consistent with their
hypothesis that successive shots are like coin tosses. They carried out this
hypothesis test for each of the 9 players in Table 2, and for all the
players combined. Here's what they found for all the players combined:
({There's more work to be done once this table is constructed; the actual
test needs to be carried out. They used something called a `chi-squared
test'. More later.})

                high    medium  low
observed        33.5%   39.4%   27.1%
expected        34.4%   36.8%   28.8%

It might seem odd to use as a hypothesis that there is no streak shooting,
rather than, for example, to hypothesize that there is streak shooting. It
is very common to test such `non-event' hypotheses; so common that the
hypothesis being tested is called the null hypothesis. There are two reasons
for this. The first reason is pragmatic: it's pretty simple to construct a
probability model for the no streak shooting assumption (coin tossing). It's
quite a bit more difficult to construct a probability model for streak
shooting. The second reason is caution. Until the data provide strong
evidence that an alternative explanation is needed, it is usually wiser to
stay with the simpler and better understood explanation.

If researchers have a particular alternative model in mind, that is
undoubtedly the correct model if the date are not consistent with the null
hypothesis, they are in a much stronger position. They can then see whether
the data provide stronger evidence for the null hypothesis model or the
alternative hypothesis. What is more often the case is that a very well
specified alternative model is not available, but a general idea about some
characteristics of possible alternative models are available. In this case,
statisticians will usually try to test the null hypothesis in a way that is
particularly sensitive to these characteristics. The technical term for the
sensitivity of an hypothesis test is ``the power of the test''.

Reference

TG
     The cold facts about the ``hot hand'' in basketball. A. Tversky and T.
     Gilovich (1989) Chance 2 (1) 16--21. (handout on November 7 for
     November 14)
