First, we learned the material found in the above classroom notes. Then using mixed references, we elegantly calculated the empirical median, empirical lower and upper quartiles, minimum and maximum of day 2 and day 4 cholesterol (see the first workbook of the above classroom task).
Then we learned the material of the above blackboard photo. From a general point of view, the Z test can be summarized in the following way. We calculate the Z statistic. It is a random variable since it is a function of the sample. Under the null hypothesis, we know its distribution (standard normal). Using this fact, we choose an interval for which it is true that under the null hypothesis the probability that Z lies in the interval equals 1-Epsilon, where Epsilon is the prescribed probability of type I error. We choose the interval to be as good as possible for the type II error probabilities (the probabilities differ according to which element of the alternative hypothesis is considered), but we do not control exactly these probabilities. This latter sentence means in our case (the test learned is two-sided) that we choose the interval to be symmetric around 0. The decision rule of the test is the following: if Z lies in the interval then we accept the null hypothesis otherwise we reject it. I draw your attention to the following essential facts: (i) In case of fixed sample size if we decrease the type I error probability then the type II error probabilities increase (ii) In case of fixed type I error probability for each element of the alternative hypothesis the type II error probability goes to 0, i.e., the test is consistent. After that we calculated the type II error probabilities explicitly.
Then we simulated the type II error probability in case of n=10, Epsilon=0.1, sigma=7, m0=180, m=175. I emphasized that the idea of simulation is quite important since the theoretical calculation is frequently not possible. Finally we plotted the theoretical type II error probability as a function of the true expectation m.
