A Day on the Dock Two crews work on a receiving dock at a fabric manufacturing plant. The first crew unloads four shipments every day and the second crew unloads seven shipments every day. A supervisor records whether each shipment is complete (a success) or missing items (a failure). Suppose X1 is a binomial random variable, representing the number of complete shipments for crew 1, with parameters n1 = 4 and p = 0.6. Similarly, let X2 be a binomial random variable, representing the number of complete shipments for crew 2, with parameters n2 = 7 and p = 0.6. Assume X1 and X2 are independent.
a. Use technology to generate a random observation for X1 (the number of complete shipments for crew 1) and a random observation for X2 (the number of complete shipments for crew 2). Add these two values to compute a random total number of complete shipments for crews 1 and 2. Repeat this process to generate 1000 random total number of complete shipments for crews 1 and 2. Compute the relative frequency of occurrence of each observation. Suppose Y is a binomial random variable with n = 11 and p = 0.6. Use technology to construct a table of probabilities for Y = 0, 1, 2, 3, c, 11. Compare these probabilities with the relative frequencies obtained above.
b. Suppose a new receiving crew is added and it unloads five shipments each day. Let X3 be a binomial random variable, representing the number of complete shipments for crew 3, with parameters n3 = 5 and p 5 0.6. Use technology to generate random observations for X1, X2, and X3. Add these three values to compute a random total number of complete shipments for crews 1, 2, and 3. Repeat this process to generate 1000 random total number of complete shipments for crews 1, 2, and 3. Compute the relative frequency of occurrence of each observation. Suppose Y is a binomial random variable with n 5 16 and p 5 0.6. Use technology to construct a table of probabilities for Y = 0, 1, 2, 3, c, 16. Compare these probabilities with the relative frequencies obtained above.
c. Suppose another receiving crew is added and it unloads nine shipments each day. Let X4 be a binomial random variable, representing the number of complete shipments for crew 4, with parameters n2 = 9 and p = 0.6. Let Y represent the total number of complete shipments for all four crews.
i. Find P(Y = 15) (the probability of exactly 15 total complete shipments).
ii. Find P(Y ≤ 12).
iii. Find P(Y > 16).
iv. How many total complete shipments can be expected?