Hypothesis Testing Power Curves #35

Create the content for the power curves section:

In this section we will explore power curves by plotting a bunch of them. We start in the situation of a z-test (known variance) by way of example. Define the power curve as a function of mu_1-mu_0, sigma, n and alpha. The dependence of the first two is continuous, for n we obviously take steps of 1 and for alpha we can take steps 0.1, 0.05, 0.025, 0.01, 0.005. Best would be to have a different color curve for each significance level as a function of mu_1-mu_0 and then have sigma and n on sliders. We want to show this separately for left-sided, right-sided and two-sided test. A short note on the t-test setting where sigma is of course not know, but we may estimate it using sample standard deviation of a pre-sample or of the sample itself. Then we also cover a different test. We can do a hypothesis test where the model distribution is Poisson distributed and we use the sum (also Poisson distributed) as the test statistic. The power will now be dependent on mu_1-mu_0, mu_1 (since variance is nmu_1), n and alpha. Again we will want multiple curves (one for each alpha). We want to leave mu_1 and n as sliders. And again we can do left-sided, right-sided and two sided. Keep in mind however that mu_0 >0 so the range fort mu_1-mu_0 may need to be adapted for some cases at some values for mu_1.

Bonus: in the slides you can find graphs of overlapping pdfs. The pdf of the test statistic under H_0 and the pdf of the test statistic under H_1. Then the type I and type II error probabilities can be indicated using shading. We can maybe recreate these also with sliders for n, sigma, alpha, and mu_1-mu_0 for the two examples above.