Below is the data from a normal distribution with mean 0 and variance \(\theta\).

-3.79 -2.32 -2.3 -2.1 -2.03 -1.37 -1.33 -1.04 -0.76 -0.75 -0.68 -0.68 -0.67 -0.64 -0.57 -0.52 -0.5 -0.47 -0.45 -0.44 -0.4 -0.39 -0.38 -0.26 -0.24 -0.24 -0.23 -0.15 -0.12 -0.09 -0.01 0.01 0.04 0.05 0.08 0.16 0.17 0.27 0.5 0.66 0.71 0.77 0.78 0.87 1.07 1.31 1.9 2.52 2.69 2.94

We want to test

\[H_0:\theta=\theta_0\text{ vs }H_a:\theta>\theta_0\]

  1. Find the likelihood ratio test based on the \(\chi^2\) approximation and apply it to the data with \(\theta_0=1,\alpha=0.05\) by finding the p-value.

  2. Do the same test as in a, but now without the large sample approximation.

  3. Draw the power curve for this test, with \(n=50, \alpha=0.05\).

  4. Derive a test based on the mle and the central limit theorem.

  5. Which test is more powerful? Add the power curve of this test to the graph in c.