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\title{Tests of statistical significance}
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\author{Carl Mason\\carlm@demog.berkeley.edu}



\section{Introduction}
\label{sec:Intro}

A test of statistical significance is a procedure designed to
determine how likely it is that the forces of total randomness could
produce a certain observed outcome that the scientist \emph{might prefer} to
attribute to a hypothetical mechanism.

For example, we might measure the heights of two sets of
women. Suppose each sample set contains 25 women and group A has an
average height of 167cm while group B has a height of 170cm. It is
possible that randomness alone would produce a difference of 3 cm in two
samples of 25 drawn from \emph{the same population/universe}.  It is
also possible that group B is taller on average than group A because
of some underlying difference between the group members. Maybe one
group lives in a warmer climate... perhaps one group has taller
mothers .. or perhaps we know nothing about how the groups were
selected and wish only to infer something about the likelihood of an
unobserved underlying difference.

A test of statistical significance does not produce a statement of
fact or certainty but rather produces a test statistic like .0992. This test
statistic can generally be taken to mean: \emph{the probability that
  chance alone could produce this outcome is .0992} or equivalently,
9,008 times out of 10,000 -- if chance alone were operating -- the
observed outcome would \textbf{not} have happened.

Hmmm does that mean that chance is responsible ?



\section{Excercise}

Consider the following chunk of code:

x1 <- rnorm(n=10, mean = 100, sd=100)
x2 <- rnorm(n=10, mean = 100, sd=100)
mean(x1);mean(x2)


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