From Algorithms to Z-Scores - matloff - University of California, Davis

15.19. CASE STUDIES 315
Exercises
Note to instructor: See the Preface for a list of sources of real data on which exercises can be
assigned to complement the theoretical exercises below.
1. In the quartic model in ALOHA simulation example, find an approximate 95% confidence
interval for the true population mean wait if our backoff parameter b is set to 0.6.
Hint: You will need to use the fact that a linear combination of the components of a multivariate
normal random vector has a univariate normal distributions as discussed in Section 8.5.2.1.
2. Consider the linear regression model with one predictor, i.e. r = 1. Let Yi and Xi represent the
values of the response and predictor variables for the i th observation in our sample.
(a) Assume as in Section 15.12.3 that V ar(Y |X = t) is a constant in t, σ 2 . Find the exact value
of Cov( ˆ β0, ˆ β1), as a function of the Xi and σ 2 . Your final answer should be in scalar, i.e.
non-matrix form.
(b) Suppose we wish to fit the model mY ;X(t) = β1t, i.e. the usual linear model but without the
constant term, β0. Derive a formula for the least-squares estimate of β1.
3. Suppose the random pair (X, Y ) has density 8st on 0 < t < s < 1. Find mY ;X(s) and
V ar(Y |X = t), 0 < s < 1.
4. The code below reads in a file, data.txt, with the header record
"age", "weight", "systolic blood pressure", "height"
and then does the regression analysis.
Suppose we wish to estimate β in the model
Fill in the blanks in the code:
mean weight = β0 + β1height + β2age
dt