This morning, in our mathematical statistical class, we’ve seen the use of the chi-square test. The first one was related to some goodness of fit of a multinomial distribution. Assume that . In order to test against , use the statistic
Under , . For instance, we have the number of weddings, in a large city, per season,
We want to test if weddings are celebrated uniformely over the year, i.e. .
> np=rep(sum(n)/4,4) > cbind(n,np) n np [1,] 301 333 [2,] 356 333 [3,] 413 333 [4,] 262 333 > Q=sum( (n-np)^2/np ) > Q  39.02102
This quantity should be compared with the quantile of the chi-square distribution
> qchisq(.95,df=4-1)  7.814728
but it is also possible to compute the p-value,
> 1-pchisq(Q,df=4-1)  1.717959e-08
Here, we reject the assumption that weddings are celebrated uniformly over the year.