15.12 Learning math through subliminal messages. A “subliminal” message is below our threshold of awareness but may, nonetheless, influence us. Can subliminal messages help students learn math? A group of students who had failed the mathematics part of the City University of New York Skills Assessment Test agreed to participate in a study to find out. All received a daily subliminal message, flashed on a screen too rapidly to be consciously read. The treatment group of 10 students was exposed to the message “Each day I am getting better in math.” The control group of eight students was exposed to the neutral message “People are walking on the street.” All students participated in a summer program designed to raise their math skills, and all took the assessment test again at the end of the program. Here are data on the subjects’ scores before and after the program:

Treatment group Control group Pretest Posttest Pretest Posttest 18 24 18 29 18 25 24 29 21 33 20 24 18 29 18 26 18 33 24 38 20 36 22 27 23 34 15 22 7

23 36 19 31 21 34 17 27

a. The study design was a randomized comparative experiment. Outline this design.

b. Compare the gain in scores in the two groups, using a graph and numerical descriptions. Does it appear that the treatment group’s scores rose more than the scores for the control group?

c. Apply the Wilcoxon rank sum test to the posttest versus pretest differences. Note that there are some ties. What do you conclude?