I. Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system. For example, most sweet oranges grow on trees grafted to the root of a sour orange variety. Suppose each graft fails independently with probability 0.3. Five grafts are scheduled to be performed next week. Let X denote the number of grafts that will fail next week. (a) The random variable X is (choose one) (i) binomial (ii) hypergeometric (iii) negative bino- mial (iv) Poisson. (b) Give the sample space and PMF of X. (c) Give the expected value and variance of X. (d) Suppose that the cost of each failed graft is $9.00. Find: (i) The probability that the cost from failed grafts will exceed $20.00. (ii) The expected value and the variance of the cost from failed grafts. 2. Suppose that 30% of all drivers stop at an intersection having flashing red lights when no other cars are visible. Of 15 randomly selected drivers coming to an intersection under these conditions, let X denote the number of those who stop. (a) The random variable X is (choose one) (i) binomial (ii) hypergeometric (iii) negative bino- mial (iv) Poisson. (b) Give the expected value and variance of X. (c) Find the probabilities P(X = 6) and P(X l: 6). You may use R commands. 4. A test consists of 10 true-false questions. Suppose a student answers the questions by ipping a coin. Let X denote the number of correctly answered questions. (a) Give the expected value and variance of X. (b) Find the probability the student will answer correctly exactly 5 of the questions. (c) Find the probability the student will answer correctly at most 5 of the questions. Use the CDF to answer this question. (d) Let Y = 10 X. In words, what does Y represent? (e) Use the CDF to find P(2 5 Y 5 5)