2. Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is N(μ_x, σ_x^2) and Y is N(μ_Y, σ_Y^2). A sample of size $n=25$ yielded x=33.80, s_x^2=4.88, and a sample of size $m=30$ yielded y=31.66, s_Y^2=5.81. Test (a) (10%)H0: σ_x^2 / σ_Y^2 = 1 against a two-side alternative with α = 0.02.

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

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