A team of psychologists studied the concept of visualization in basketball, where players visualize making a basket before shooting the ball. They conducted an experiment in which 20 basketball players with similar abilities were randomly assigned to two groups. The 10 players in group 1 received visualization training, and the 10 players in group 2 did not. Each player stood 22 feet from the basket at the same location on the basketball court. Each player was then instructed to attempt to make the basket until two consecutive baskets were made. The players who received visualization training were instructed to use visualization techniques before attempting to make the basket. The total number of attempts, including the last two attempts, were recorded for each player. (a) Based on the boxplots, did basketball players who received visualization training tend to need fewer attempts to make two consecutive baskets from a distance of 22 feet than players who did not receive the training? Explain your reasoning.

(b) State and check conditions for conducting a two-sample t -test for a difference in means.

Because both distributions, visualization (V) and no visualization (NV) , are skewed, the psychologists conducted a simulation to test for a difference in medians rather than means. For each trial of the simulation, the 20 values of the total number of attempts observed in the experiment were combined into one group and then randomly split into two groups of 10. The difference in the medians (V−NV) of the groups was calculated for each trial. The following dotplot shows the difference in the medians for 100 trials of the simulation.

c) Using the observed difference in medians (V−NV) and the results of the simulation, estimate a p -value for a test for the difference in medians. Show the work needed to calculate this p -value.

(d) Based on the p -value in part (c), is there convincing statistical evidence that basketball players similar to the ones in this study who receive visualization training need fewer attempts to make two consecutive baskets from a distance of 22 feet than those who do not receive such training? Justify your answer.