A hypothesis test at the 5% significance level leads to the rejection of $H_0$. Explain why this does not prove that $H_1$ is true.

Explain why lowering the significance level from 5% to 1% makes it harder to reject $H_0$.

For a two-tailed test on a binomial proportion at a nominal 5% significance level, the critical region chosen has an actual significance level of 2.97%. Explain why the two figures differ, and state the consequence for the test.