In Week 7 we saw how to compare two population proportions, 
and 
. In this section we consider proportions for more than
two 0-1 populations. If we have K such populations and
we have random samples
of size 
from the populations, then typically we
want to test whether whether the true population proportions
are some hypothesized values 
. The most common example
is whether all proportions are the same.
If the null hypothesis is true, then we would expect to get
1's (that is `successes') in the ith sample. If we let
denote the actual number of 1's in the ith sample, then
we could measure how far the observed data is from what we expect
if the null hypothesis is true by test statistic
A large (small) value of this statistic is evidence against (not against) the null hypothesis.
We reject the null hypothesis if 
.