# Applications of Chi-Square Tests

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 $\boldsymbol{N}=(N_1,\cdots,N_k)\sim\mathcal{M}(n,\boldsymbol{p})$. In order to test $H_0:\boldsymbol{p}=\boldsymbol{p}_0$ against $H_1:\boldsymbol{p}\neq\boldsymbol{p}_0$, use the statistic

$Q=\sum_{j=1}^k \frac{[N_j-np_{0,j}]^2}{np_{0,j}}$Under $H_0$$Q\sim\chi^2(k-1)$. For instance, we have the number of weddings, in a large city, per season,

> n=c(301,356,413,262)

We want to test if weddings are celebrated uniformely over the year, i.e. $H_0:\boldsymbol{p}=\boldsymbol{1}/4$.

> 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
[1] 39.02102

This quantity should be compared with the quantile of the chi-square distribution

> qchisq(.95,df=4-1)
[1] 7.814728

but it is also possible to compute the p-value,

> 1-pchisq(Q,df=4-1)
[1] 1.717959e-08

Here, we reject the assumption that weddings are celebrated uniformly over the year.