An examination was given to 500 high-school students in each of two large cities, and their grades were recorded as low, medium, or high. The results are given in the table below.

\begin{tabular}{l|ccc} & Low & Medium & High \ \hline City A & 103 & 145 & 252 \ City B & 140 & 136 & 224 \end{tabular}

Derive carefully the test of homogeneity and test the hypothesis that the distributions of scores among students in the two cities are the same.