(Some critical values $F_\alpha(df_1, df_2)$ and $t_\alpha(df)$,

where $\Pr(F > F_\alpha(df_1, df_2)) = \alpha$ and $\Pr(t > t_\alpha(df)) = \alpha$, are given as follows:

$$
\begin{array}{l}
F_{0.02}(24, 29) = 2.236, F_{0.01}(24, 29) = 2.495, F_{0.02}(29, 24) = 2.306, \\
F_{0.01}(29, 24) = 2.589, F_{0.02}(30, 25) = 2.267, F_{0.01}(30, 25) = 2.538, \\
t_{0.01}(53) = 2.398, t_{0.01}(54) = 2.397, t_{0.01}(55) = 2.396, \\
t_{0.005}(53) = 2.672, t_{0.005}(54) = 2.670, t_{0.005}(55) = 2.668
\end{array}
$$

3. Let $X_1, X_2, \ldots$, and $X_n$ be a random sample from distributions with the following probability density functions. In each case find the maximum likelihood estimator $\hat{\theta}$.

(a) \((5\%) f(x; \theta) = \theta x^{\theta - 1}, 0 < x < 1, 0 < \theta < \infty\)