Homework assignment, due mardi, 27 oct

1. Exercise 7.1: Discrete probability simulation: suppose that a basketball player has a 60% chance of making a shot, and he keeps taking shots until he misses two in a row. Also assume his shots are independent (so that each shot has 60% probability of success, no matter what happened before).

(a) Write an R function to simulate this process.

(b) Put the R function in a loop to simulate the process 1000 times. Use the simulation to estimate the mean, standard deviation, and distribution of the total number of shots that the player will take.

(c) Using your simulations, make a scatterplot of the number of shots the player will take and the proportion of shots that are successes.

2. Exercise 7.3: Propagation of uncertainty: we use a highly idealized setting to illustrate the use of simulations in combining uncertainties. Suppose a company changes its technology for widget production, and a study estimates the cost savings at $5 per unit, but with a standard error of $4. Furthermore, a forecast estimates the size of the market (that is, the number of widgets that will be sold) at 40,000, with a standard error of 10,000. Assuming these two sources of uncertainty are independent, use simulation to estimate the total amount of money saved by the new product (that is, savings per unit, multiplied by size of the market).

3. Exercise 7.8: Inference for the ratio of parameters: a (hypothetical) study compares the costs and effectiveness of two different medical treatments.

• In the first part of the study, the difference in costs between treatments A and B is estimated at $600 per patient, with a standard error of $400, based on a regression with 50 degrees of freedom.

• In the second part of the study, the difference in effectiveness is estimated at 3.0 (on some relevant measure), with a standard error of 1.0, based on a regression with 100 degrees of freedom.

• For simplicity, assume that the data from the two parts of the study were collected independently. Inference is desired for the incremental cost-effectiveness ratio: the difference between the average costs of the two treatments, divided by the difference between their average effectiveness. (This problem is discussed further by Heitjan, Moskowitz, and Whang, 1999.)

(a) Create 1000 simulation draws of the cost difference and the effectiveness difference, and make a scatterplot of these draws.

(b) Use simulation to come up with an estimate, 50% interval, and 95% interval for the incremental cost-effectiveness ratio.

(c) Repeat this problem, changing the standard error on the difference in effectiveness to 2.0.