A pathological video game user (PVGU) is a video game user that averages 31 or more hours a week of game-play. According to the article "Pathological Video Game Use among Youths: A Two-Year Longitudinal Study" (Pediatrics, Vol. 127, No. 2, pp. 319-329) by D. Gentile et al., in 2011, about 9% of children in grades 3-8 were PVGUs. Suppose that, today, seven youths in grades 3-8 are randomly selected.

a. Assuming that the percentage of PVGUs in grades 3-8 is the same today as it was in 2011, determine the probability distribution for the number, X, who are PVGUs.

b. Determine and interpret the mean of X.

c. If, in fact, exactly three of the seven youths selected are PVGUs, would you be inclined to conclude that the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning. Hint: First consider the probability P(X ≥ 3).

d. If, in fact, exactly two of the seven youths selected are PVGUs, would you be inclined to conclude that the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning.