The most recent data from the Department of Education show that 34.8% of students who submitted otherwise valid applications for a Title IV Pell Grant in 2005–2006 were ineligible to receive such a grant (www2.ed.gov/finaid/prof/resources/data/pell-2005-06/eoy-05-06.pdf). Suppose that this result is true for the current population of students who submitted otherwise valid applications for this grant.
a. Let x be a binomial random variable that denotes the number of students in a random sample of 20 who submitted otherwise valid applications for a Title IV Pell Grant but were ineligible to receive one. What are the possible values that x can assume?
b. Find the probability that exactly 6 students are ineligible to receive a Title IV Pell Grant in a random sample of 20 who submitted otherwise valid applications for this grant. Use the binomial probability distribution formula.