Introduction to Statistical Thinking

In this exercise we would like to analyze the results of the trial that involves magnets as a treatment for pain. The trial is described in Question [ex:Inference.1]. The results of the trial are provided in the file magnets.csv.

Patients in this trail where randomly assigned to a treatment or to a control. The responses relevant for this analysis are either the variable change, which measures the difference in the score of pain reported by the patients before and after the treatment, or the variable score1, which measures the score of pain before a device is applied. The explanatory variable is the factor active. This factor has two levels, level 1 to indicate the application of an active magnet and level 2 to indicate the application of an inactive placebo.

In the following questions you are required to carry out tests of hypotheses. All tests should conducted at the 5% significance level:

It is assumed, when constructing the \(F\)-test for equality of variances, that the measurements are Normally distributed. In this exercise we want to examine the robustness of the test to divergence from the assumption. You are required to compute the significance level of a two-sided \(F\)-test of \(H_0:\Var(X_a)=\Var(X_b)\) versus \(H_1: \Var(X_a)\not =\Var(X_b)\text{.}\) Assume there are \(n_a=29\) observations in one group and \(n_b = 21\) observations in the other group. Use an \(F\)-test with a nominal 5% significance level.

The sample average in one sub-sample is \(\bar x_a = 124.3\) and the sample standard deviation is \(s_a = 13.4\text{.}\) The sample average in the second sub-sample is \(\bar x_b = 80.5\) and the sample standard deviation is \(s_b = 16.7\text{.}\) The size of the first sub-sample is \(n_a=15\) and this is also the size of the second sub-sample. We are interested in the estimation of the ratio of variances \(\Var(X_a)/\Var(X_b)\text{.}\)