Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether goodgrades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. Supposewe select one of the students at random.MostimportantGradesAthleticPopular4thgrade492419Total 92Grade5thgrade5036221086thgrade693828135Total1689869335Are the events "5th grade" and "athletic" independent?Justify your answer.P(athletic) = = 0.293 and36P(athletic | 5th grade) == 0.3333618+18335Because these two probabilities are not equal, theevents "athletic" and "5th grade" are independent.P(athletic and 5th grade) = 3 = 0.107 andP(athletic and not 5th grade) = = 0.107Because these two probabilities are equal, the events"athletic" and "5th grade" are independent.P(athletic and 5th grade) = = 0.107 andP(athletic and not 5th grade) = = 0.185Because these two probabilities are not equal, the3633524+38335events "athletic" and "5th grade" are not98335independent.P(athletic) = 98 = 0.293 and335P(athletic | 5th grade) = 36 = 0.333108Because these two probabilities are not equal, theevents "athletic" and "5th grade" are notindependent.98P(athletic) = 335 = 0.293 and36P(athletic | 5th grade) = 3 = 0.293335Because these two probabilities are not equal, theevents "athletic" and "5th grade" are notindependent.

Given data: The objective is to determine whether the events "5th grade" and "athletic" independent. Two events A and B are said to be independent if PA and B=PA×PB. Pathletic=98335≈0.293 P5th grade=108335≈0.3224 This implies: Pathletic×P5th grade=98335×108335=0.2925×0.3224≈0.0943 The probability that a randomly selected student is an athletic given that he is in the 5th grade is: Pathletic | 5th grade=Pathletic ∩ 5th gradeP5th grade=36335108335=36108≈0.3333 This implies: Pathletic ∩ 5th grade=Pathletic | 5th grade×P5th grade=36108×108335=0.3333×0.3224≈0.1074 Since Pathletic×P5th grade≠Pathletic and 5th grade, the two events are not independent. OR Pathletic=98335≈0.293 Pathletic | 5th grade=Pathletic ∩ 5th gradeP5th grade=36335108335=36108≈0.333 Because these probabilities are not equal, the events 'athletic' and '5th grade' are not independent. Option (d) is correct.