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167 STATISTICS
numbers do not add up to 100 per cent because there are other possibilities).
Thus 17 per cent is our best guess at this proportion among the whole population
of written restrictive relative clauses. However, since we have only taken a
sample of the population our result must be expressed tentatively, acknowledging
the uncertainty introduced by examining only a subset of the population
rather than taking a census (in this particular case, a census would not even
be possible). So rather than reporting a simple value, statisticians often prefer
to quote a confidence interval (i.e. a range of values within which we are reasonably
confident the true population value lies).
Statistical inference
When you have collected your data, and made some estimates of various quantities
of interest, you may simply wish to report those estimates, with their
uncertainties. Most commonly, however, you want to go a step further and
answer a specific research question: e.g. do the speakers in two different
suburbs really behave in a different way? In particular do the mean values of a
particular vowel differ? You will have collected samples from both suburbs and
calculated means in the two samples. Those sample means will undoubtedly
differ, at least slightly. But does that mean that the population means are
different? You will probably have individuals in each sample whose behaviour
is atypical of the population they came from, for reasons such as those suggested
above, but you want to know whether the differences between the two
samples are due entirely to chance (because of the individuals you recorded
when trying to find out the answer to your question) or whether the two populations
they came from are really behaving differently. Statisticians prefer us
to ask this question in a very particular way. Let us assume, they say, that there
is no difference between our two populations (i.e. the people from the different
suburbs, the very tall and not-so-tall people, are in principle behaving identically).
This is the null hypothesis. It is, in a sense, the hypothesis of the
person who is sceptical about your expectations and believes that the two
groups do not differ in terms of the feature we are trying to measure. Statistical
tests are then framed so as to answer the question: what evidence do we have
that the null hypothesis is wrong? In a statistical test we start from the null
hypothesis, and look to see if we have evidence to persuade us out of that
position.
p-values
At this stage we have stopped asking about the nature of the sample, and we are
trying to use the sample to draw inferences about the population as a whole.
The null hypothesis is that the two samples we are dealing with really belong