Quick questions on Hypothesis testing and errors explained: H2 Further Mathematics

A hypothesis test sets a null hypothesis $H_0$ and an alternative $H_1$, chooses a significance level $\alpha$, defines a rejection region, and rejects $H_0$ if the test statistic falls in that region. The decision can be correct or can commit one of two errors.

By construction, the significance level $\alpha$ is the probability of a Type I error, because the rejection region is chosen so that, when $H_0$ is true, the test statistic lands there with probability $\alpha$. Choosing a smaller $\alpha$ reduces Type I errors but, for fixed sample size, increases Type II errors.

The power is the probability of correctly rejecting a false $H_0$:

Define a Type I error. [1 mark]

What is the power of a test in terms of $\beta$? [1 mark]

Why does reducing $\alpha$ tend to increase $\beta$ for a fixed sample size? [2 marks]