Introduction to Statistical Thinking

Consider a medical condition that does not have a standard treatment. The recommended design of a clinical trial for a new treatment to such condition involves using a placebo treatment as a control. A placebo treatment is a treatment that externally looks identical to the actual treatment but, in reality, it does not have the active ingredients. The reason for using placebo for control is the "placebo effect". Patients tend to react to the fact that they are being treated regardless of the actual beneficial effect of the treatment.

As an example, consider the trial for testing magnets as a treatment for pain. The patients that were randomly assigned to the control (the last 21 observations in the file "magnets.csv") were treated with devices that looked like magnets but actually were not. The goal in this exercise is to test for the presence of a placebo effect using the data in the file "magnets.csv".

It is assumed, when constructing the \(t\)-test, that the measurements are Normally distributed. In this exercise we examine the robustness of the test to divergence from the assumption. You are required to compute the significance level of a two-sided \(t\)-test of \(H_0:\Expec(X)=4\) versus \(H_1: \Expec(X) \not = 4\text{.}\) Assume there are \(n=20\) observations and use a \(t\)-test with a nominal 5% significance level.

Assume that you are interested in testing \(H_0:\Expec(X) = 20\) versus \(H_1:\Expec(X)\not = 20\) with a significance level of 5% using the \(t\)-test. Let the sample average, of a sample of size \(n=55\text{,}\) be equal to \(\bar x = 22.7\) and the sample standard deviation be equal to \(s = 5.4\text{.}\)