Define a simple hypothesis. Define the terms size and power for a test of one simple hypothesis against another. State the Neyman-Pearson lemma.

There is a single observation of a random variable $X$ which has a probability density function $f(x)$. Construct a best test of size $0.05$ for the null hypothesis

$$H_{0}: \quad f(x)=\frac{1}{2}, \quad-1 \leqslant x \leqslant 1,$$

against the alternative hypothesis

$$H_{1}: \quad f(x)=\frac{3}{4}\left(1-x^{2}\right), \quad-1 \leqslant x \leqslant 1 .$$

Calculate the power of your test.