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.

grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. Suppos we select one of the students at random. Most important 4th grade Grades 49 Athletic 24 Popular 19 Total 92 Grade 5th 6th grade grade 50 69 36 38 22 28 108 135 Total 168 98 69 335 Are the events "5th grade" and "athletic" independent? Justify your answer. 98 P(athletic) = 335 = 0.293 and P(athletic | 5th grade) = = 0.333 Because these two probabilities are not equal, the events "athletic" and "5th grade" are independent. P(athletic and 5th grade) = 3 = 0.107 and