Two professors at the University of Arizona were interested in whether having students actually play a game would help them analyze theoretical properties of the game. The professors performed an experiment in which students played one of two games before coming to a class where both games were discussed. Students were randomly assigned to which of the two games they played, which we’ll call Game 1 and Game 2. On a later exam, students were asked to solve problems involving both games, with Question 1 referring to Game 1 and Question 2 referring to Game 2. When comparing the performance of the two groups on the exam question related to Game 1, they suspected that the mean for students who had played Game 1 (μ₁) would be higher than the mean for the other students μ₂, so they considered the hypotheses H₀: μ₁ = μ₂ vs H_a: μ₁ > μ₂.

(a) The paper states: ‘‘test of difference in means results in a p-value of 0.7619.” Do you think this provides sufficient evidence to conclude that playing Game 1 helped student performance on that exam question? Explain.

(b) If they were to repeat this experiment 1000 times, and there really is no effect from playing the game, roughly how many times would you expect the results to be as extreme as those observed in the actual study?

(c) When testing a difference in mean performance between the two groups on exam Question 2 related to Game 2 (so now the alternative is reversed to be H_a: μ₁ < μ₂ where μ₁ and μ₂ represent the mean on Question 2 for the respective groups), they computed a p-value of 0.5490. Explain what it means (in the context of this problem) for both p-values to be greater than 0.5.