A group of researchers has estimated that the probability with which a teacher retires in any given year is 0.3. (a) (2 points) A school currently has 5 teachers in third grade. What is the probability that 3 of them will retire next year? (b) (3 points) A school currently has 5 teachers in third grade. What is the probability that most of them will retire next year? (c) (4 points) A school district has 500 teachers. What is the probability that more than 160 of them will retire next year? (d) (4 points) What is the probability that a given teacher has retired within the next three years? 2. The government introduces a policy providing incentives for teachers to retire earlier. As a result the probability with which a teacher retires in a given year might change from its previous value of 0.3. (a) (6 points) A group of researchers has observed 320 teachers after the policy of early retirement incentives was implemented. 112 of them decided to retire. Test whether the probability of a teacher retiring after was implemented is now larger than 0.3, using a 5% significance level. (b) (4 points) Using the sample information above, calculate a 99% confidence interval for the new probability with which a teacher retires in a given year.