Two new mathematics learning techniques are being tested. twenty students were randomly selected from a population. na = 9 of them were randomly assigned to use technique a, and nb = 11 of them were randomly assigned to use technique b. each student spent 30 minutes learning the technique to which they were assigned, and then were asked to complete a task. the time to complete the task was recorded, in seconds. a shorter time indicates better mastery of the task. the data are below: technique a: 23.1, 21.4, 31.6, 34.5, 21.9, 36.0, 30.2, 33.1, 39.5 technique b: 32.7, 36.8, 39.1, 37.3, 40.3, 46.8, 41.4, 53.0, 55.6, 54.1, 28.3 we wish to test: h0 : µa − µb = 0 vs. ha : µa − µb 6= 0 using α = 0.05.

(a) graph the data as you see fit. why did you choose the graph(s) you did and what does it (do they) tell you? also calculate summary statistics relevant to the research question.

(b) use the bootstrap-t test to test h0, using b = 1 resamples. display the histogram of your generated t ∗ values. compute your tobs and a p-value. make a reject or not reject decision. finally, state your conclusion in the context of the problem.