a) P(at least 16 years old) = P(16 years old) + P(17 years old) + P(18 years old) + P(19 years old) P(at least 16 years old) = 0.211+0.321+0.151+0.084 = 0.767
There is a 0.767 probability that if a student athlete is at least 16 years old if chosen from the district at random.
b) E(x) = mux = ∑xi * Pi = (140.094)+(150.139)+(160.211)+(170.321)+(180.151)+(190.084) = 16.548 years
The expected age of a randomly-selected student-athlete is 16.548 years.
c) Binomial Distribution Conditions:
Binary - at least 16 or under 16, Independent - since you can only be in no more than one age group, the student-athletes’ ages are independent, Number of Trials = there are five students who are selected (n=5), Same Probability = probability of being at least 16 is the same for each student-athlete (p = 0.767)
X = the number of student-athletes that are at least sixteen years old where the distribution of x is binomial with p = 0.767 and n = 5
Using binomial cdf on TI Calculator with n = 5, p = 0.767, lower bound = 4, upper bound = 5, the calculated probability is 0.6686.
There is a 0.6686 probability that 4 or 5 of the selected athletes are at least 16 years old if a random sample of 5 student-athletes is selected.