You're wrong on two accounts.
You want a low p-value, the lower the p-value the stronger the evidence.
A p-value is cutoff traditionally at .05 (1 in 20), and if it's p < .05 you can reject the null hypothesis. The "null hypothesis" is the OPPOSITE of your paper's hypothesis (so you want to reject it): the null hypothesis is "All the variation we saw was from chance"
