In this lecture we're going to continue
our talk about hypothesis testing of
Gaussian distributed data. So to
elaborate on how hypothesis tests work
we're going to talk about the hypotheses
a little bit more. So there are two
hypotheses when we're doing a
statistical test. There's a null
hypothesis which we call h-naught, and it
usually represents the control data or
random data that results purely from
chance. The second hypothesis is the
alternative hypothesis. So this is the
thing we're trying to prove, generally.
For example, if we're doing an experiment
where we're testing a drug, the drug
actually working would be the
alternative hypothesis. And so the
alternative hypothesis is the hypothesis
that sample observations are influenced
by some non-random cause. So now suppose
we have some random data, so let's say R1
equals randn. Let's say it as a hundred points,
a mean of zero, and a variance of one.
Now let's say we have a another data set
which is maybe 20 points,
but this one's going to have a different
mean. So let's say it's mean is one, and
let's increase its variance a little bit.
Okay, so, we have two distributions that
are Gaussian distributed. The first one
has 0 mean and one variance, and the second
one has a mean of one but a standard
deviation of two. So, how do we compare
these two distributions? There is a test
called the ttest, or the two-sided
t-test, or the two-sample t-test which does
what we want, and it again returns a
hypothesis and a p-value. So we can try
this see ttest2(R1,R2). Ok, so, we reject the
null hypothesis in this case with a very,
very small p-value. Remember it only has
to be less than five percent for us to
reject the null hypothesis. So let's try
some things. Let's
have less data points for R1. Okay, and
let's do our ttest again. So notice how
we still reject the null hypothesis but
our p value has increased, so it's less
significant than before. Alright, so, now
let's do the same thing for R2,
let's say this now only has 10 points.
Let's do the ttest again. Alright, so,
this is still significant. Let's increase
the variance.
Alright, so, I had to increase the
variance a lot to get an insignificant p
value, so that's one thing about when
you're comparing two gaussian
distributions you can't really say one
is bigger than the other if they're
spread out a lot. So let's put the
variance back for R2, but let's say
the mean is now less far away from
R1's mean. Let's do our ttest again, and
so this also gives us an insignificant p
value. So that's another fact about the
ttest is you also can't tell if two
distributions are different if they are
very close together. If they're very far
apart, so the mean is, let's say we do the
ttest again, we now get a very small
p-value.