3 1. State the geometric reason that for a dummy variable model with a single dummy variable (i.e. yi = 0 + 1 xi + ei , where xi = 1 if success, 0 if failure) that the Psuch 5 first 5 observations are successes and the last 5 are failures (n = 10), i=1 ei = 0. 2. (From Meyer, Practical Statistical Models, 2002.) A botanist is interested in the efficacy on predator bugs in reducing pests on garden plants. In particular, two species (A and B) of praying mantis are to be compared to see which devours potato beetles at a higher rate. One hundred grams of potato beetles are released into each of four cages containing potato foliage, and at the end of a week, the reduction (in grams) of potato beetles is measured. The first cage contains one mantis of each species. The second cage contains two of each species. For the third cage, there are two of Species A and one of Species B, In the fourth cage, there are two of Species B and one of Species A. Let A be the average grams of potato beetles eaten per week per praying mantis for Species A, and let B be the average for Species B. Write down a model using a matrix equation to estimate A and B , giving your design matrix X. Assume that the consumption of each praying mantis is independent of others in the cage. Don't worry about trying to solve for estimates of A and B ; just write down the model. 3. Instead of using the y-intercept as in the notes and textbook, suppose we wanted to create a linear model using two dummy variables like this one: yi = 1 x1 + 2 x2 + ei , i = 1, . . . , n. You might think of the calcium supplement - blood pressure problem from class, but this time, in general there are m people in the first group, and n m people in the second group. Our dummy variables are then defined as: \u001a x1 = 1, i = 1, . . . , m 0, i = m + 1, . . . , n \u001a x2 = 0, i = 1, . . . , m 1, m + 1, . . . , n (a) Define the parameters 1 and 2 in the context of the problem. (b) Use the usual formula = (X0 X)1 X0 y to solve for the parameter estimates of 1 and 2 . (Double-check: the estimates should be consistent with your definitions above.) 4. (From Stapleton, 1995.) Suppose we have an ordinary household scale such as might be used in a kitchen. When an object is placed on the scale, the reading is a combination of the true weight plus a random error. You have two coins of unknown weights 1 and 2 . To estimate the weights of the coins, you take four observations: 1 Put coin 1 on the scale and observe y1 . Put coin 2 on the scale and observe y2 . Put both coins on the scale and observe y3 . Put both coins on the scale again and observe y4 . Suppose the random errors are independent and identically distributed. (a) Write a linear model in matrix form and find the least-squares estimates of the coins using the usual formula (X0 X)1 X0 y. (b) Explain in words why at least parts of the estimates of the coins make sense. (The denominator 5 may not be intuitive.) 5. Question 2.5, Textbook (pp. 41-42) 6. Question 2.6, Textbook (p. 42) 7. For the simple linear regression model yi = 0 + 1 xi + ei , we find the t-statistic for testing H0 : 1 = 0 to be t = (1 1 )/se(1 ). (a) Which of the usual assumptions for the model must be met in order for the tstatistic to have the t-distribution? Why? (b) Does having a larger sample size change your answer? Why or why not? (Hint: I'm not 100% convinced the 408 students will be able to answer this question on their own; ask me about it in class.) 8. Suppose that x is a random n-dimensional vector, and that E[x] = . Show that the covariance matrix = E[(x )(x )0 ] is equal to E[xx0 ] 0 . 2