Chapter 14.4, Problem 15E

Suppose that the entries in a contingency table that appear in row i and column j are denoted n_ij, for i = 1, 2,…, r and j = 1, 2,…, c; that the row and column totals are denoted r_i, for i = 1, 2,…, r, and c_j, for j = 1, 2,…, c; and that the total sample size is n.

1. a Show that

$X^2={\displaystyle \underset{j=1}{\overset{c}{\sum }}{\displaystyle \underset{i=1}{\overset{r}{\sum }}\frac{{[n_{ij}-E({\stackrel{⌢}{n}}_{ij})]}^2}{E({\stackrel{⌢}{n}}_{ij})}}=n\left({\displaystyle \underset{j=1}{\overset{c}{\sum }}{\displaystyle \underset{i=1}{\overset{r}{\sum }}\frac{n_{ij}^2}{r_i{c}_j}-1}}\right)}.$

Notice that this formula provides a computationally more efficient way to compute the value of X².

1. b Using the preceding formula, what happens to the value of X² if every entry in the contingency table is multiplied by the same integer constant k > 0?