The standard error, or standard deviation of the mean, is given by Eq.(2.7):
$$ \alpha = \frac{\sigma_{N-1}}{\sqrt{N}}. $$To decrease $\alpha$ by a factor of 10, the denominator must be increased by the same factor, which means that $N$ must increase by a factor of 100. Translating to the described experiment, this means that data should be collected for 100 minutes (assuming that everything in the experiment is stable for that length of time).